To introduce Dharma, what categories of people are there?

Buddha's audience is not only human beings.

1 - Buddhas,
2 - Bodhisattvas
3 - "Sound-Hearing" Arhats
4 - "Enlightened by Conditions" Arhats
5 - Gods
6 - People
7 - Asuras
8 - Animals
9 - Ghosts
10 - Hells

In The Shurangama Sutra, plus one additional: 11 - Immortals

As for categories / grades of each of above, depending on how you group them.

eg There are roughly 10 big categories of people mentioned in The Shurangama Sutra, and more sub-categories.

As for Gods (excuse for my English), there are 4 in Formless realm, which is above 18 in Form realm, which again is above 6 in Desire realm: https://en.wikipedia.org/wiki/Buddhist_cosmology#Form_Realm_.28R.C5.ABpadh.C4.81tu.29

As for Bodhisattvas, there are:

10 Faiths
10 Dwellings
10 Conducts
10 Transferences
 4 Positions of Additional Practices
10 Positions of the Ten Grounds
The Position of Equal and Wonderful Enlightenment

It is difficult to group the audience s into categories as the differences are many. Each person differs in each ones' level of understanding. The skillful and unskillful qualities of a persons mind differs from person to person. Then every questioners need is different from one to another. Some questions call for analytical answers, and others require categorical answers.

The Buddha had ten special powers. One of them was the ability to know positive and negative attribute of every being. So Buddha tailored the response depending on the person’s beliefs, or need, or level of knowledge.

The examples are many. One that came to mind was that of Cula Panthaka. The Buddha helped him to remove his ‘kleshas’ & ‘upakkleshas’ and finally release his mind from all defilements. The story goes like this:

At the time of the Supreme Buddha, there were two brothers. They ordained in the Dispensation of the Buddha. The elder brother was very talented in remembering the Dhamma really well, but the second brother was unable to remember even one stanza. So, the elder brother asked his young brother to give up the monkhood and leave, “You can’t remember the Dhamma. You must go back home and start cultivating some plants.” But the younger brother didn’t want to leave the monkhood.

At the end, the younger-brother monk went to the temple’s gate and sat there crying. The Supreme Buddha came to him. “Chūla Panthaka, why are you crying?” “Oh dear Sir, my brother asked me to leave the temple.” “Why was that?” “I can’t remember any stanza I read. Therefore, he asked me to leave the monkhood and cultivate a field at home.” “No, that’s not what you should do. Your brother doesn’t own this dispensation.”

The Supreme Buddha gently touched the monk’s head and took his hand, “Come with me.” The Supreme Buddha gave this young monk a small piece of cloth. “Dear monk, rub this cloth saying ‘cleaning, cleaning (rajō haranhan)’.” The young monk did as the Buddha told him. He saw that even though his intention was to clean the cloth, it was actually getting dirty gradually, “I rubbed this saying it to clean, but it is actually getting dirty.” His mind was well concentrated on to what happened to the cloth. The Supreme Buddha preached the Dhamma, so that the monk’s mind will be cultivated. In a moment, the young monk became an Arahant. See how his talent of realizing the Dhamma was concealed within him. If it was not the Buddha himself, who else would be able to do such a miracle? How many chances we may have missed and how many talents of us realizing life may have been concealed within us? Who would know how to unveil that talent in us?

It was this Chūla Panthaka Thero who became the first among the Bhikkhus who were able to create mental (Manōmaya) bodies. His brother monk didn’t know about it. He thought Chūla Panthaka Thero has gone home by that time. On that same day, all Bhikkhus went to alms-offering invitation. The Supreme Buddha told them that one monk is missing. Nobody knew about Chūla Panthaka Thero. The Supreme Buddha asked someone to go fetch the missing monk from the temple.

When he went to the temple, he saw one thousand monks. They were all alike. They were walking, cleaning the ground, and meditating. That person who came to fetch the monk ran back to the house and told the Supreme Buddha, “Dear Sir, there are thousand monks in the temple now.” The Supreme Buddha told him, “Please go back and ask what his name is. Touch the robe of the monk whoever answers first.” So, that person went back to the temple and said, “Who among you is called Chūla Panthaka Thero?” Then all thousand monks came forward saying, “I am… I am… I am…” As soon that person touched the monk who first answered, all other nine hundred ninety nine monks (bodies) disappeared.

He said, “Dear Sir, please come to lunch.” Chūla Panthaka Thero answered, “Dear devotee, you go please. I will come later.” The Supreme Buddha had the devotees at home to make a chair for Chūla Panthaka Thero. Then a Bhikkhu appeared from beneath the ground and sat on that chair; it was the Chūla Panthaka Thero.

The Pali scriptures show two basic audiences: (i) ordinary people (puthujjana), who have or are not capable of abandoning self-view; and (ii) the noble disciple (ariyasāvaka), who has or is capable of abandoning self-view.

Thus there is a stock phrase in the Pali scriptures, namely:

...this is the difference, this the distinction, this the distinguishing factor, between a noble disciple and an ordinary person...

Thus, MN 117 lists teachings that are taken on faith by the ordinary person & teachings that are for the noble disciple:

And what is the right view with effluents, siding with merit, resulting in acquisitions? 'There is what is given, what is offered, what is sacrificed. There are fruits & results of good & bad actions. There is this world & the other worlds. There is mother & father. There are spontaneously born beings; there are contemplatives & brahmans who, faring rightly & practicing rightly, proclaim this world & the other worlds after having directly known & realized it for themselves.' This is the right view with effluents, siding with merit, resulting in acquisitions.

And what is the right view that is noble, without effluents, transcendent, a factor of the path? The discernment (wisdom), the faculty of discernment, the strength of discernment, analysis of qualities as a factor for awakening, the path factor of right view in one developing the noble path whose mind is noble, whose mind is without effluents, who is fully possessed of the noble path. This is the right view that is noble, without effluents, transcendent, a factor of the path.

For example, the teaching above states the ordinary person should believe there are 'beings' ('satta') spontaneously born due to their good & bad actions yet teachings for noble disciples, such as SN 5.10, state there are no 'beings' to be found.

This is the reason there a teachings that contradict each other, such as MN 60 which states the view of 'existence' is right view for a householder & SN 12.15 which states the view of 'existence' is wrong view for the noble disciple; or teachings such as in Sigalovada Sutta which states the duty of a monk is to point out the path to 'heaven' (rather than to Nibbana) to the lay devotee. That is why the teachings about kamma & 'rebirth' never include the teaching of 'not-self' in them.

Later scholars, such as Buddhaghosa, created ideas that impersonal not-self (empty) phenomena are reborn in cycles of rebirth but the Buddha did not teach like this.

The Pali scriptures also include some special teachings to Brahmans using their own religious ideas, such as ending 'nama-rupa' ('name-&-form') & radiating love as the path to union with Brahma (God). These teachings confuse many modern Buddhists, who take these teachings to be Buddhism rather than (improvements to) Brahmanism.

In relation to what is posted above, the scriptures never report the Buddha making the declaration that 'karma & rebirth' or 'the path to Brahma' were his personal teachings & revelations (since these basic ideas pre-existed before the Buddha or used ideas not relevant to the Buddha himself).

The scriptures (eg, AN 3.61, AN 3.134, SN 56.11, SN 56.31, SN 12.20, SN 20.7, MN 56, etc) only state the Buddha declared the 'six elements', the 'eighteen applications of mindfulness at feeling', the 'four noble truths', the 'three characteristics', 'dependent origination' & 'emptiness' to be his special & unique teachings.

Then the Blessed One discoursed to him a graduated sermon, that is to say, he spoke on the subjects of generosity, virtue, the heavens, on the evil consequences, the vanity and the depravity of sensual pleasures and on the advantages of renunciation.

When the Blessed One perceived that the mind of Upāli, the householder, was prepared, pliant, free from obstacles, elevated and lucid, then he revealed to him that special doctrine of the Buddhas, viz. Suffering, its Arising, its Ceasing and the Path.

MN 56

Kesi Sutta is where Buddha explained to a horse trainer of methods he used in teaching. It indirectly explains types of audiences.

• gentle, result of good conduct.
• harshness, result of misconduct.
• gentleness & harshness, both methods above.
• put down, Buddha regarded him as being worthless to teach.

Though I agree with other answers I'm not sure I agree with the question: i.e. I'm not sure it's sane to try to categorize the different audiences, or decide there's only one possible categorization-system (still there are many common categorization-systems, e.g. ordained or lay, enlightened or not, etc.).

What I mean is that, although it's true he spoke to groups of people (and a group might be seen as a category), sometimes he taught small groups or 'individuals': assuming each teaching was "tailored to meet the particular needs of his audience", that's a specific/individual audience (not a category).

For example, look at Ariyapariyesana Sutta (MN 26) which I think explains why he started the Dhammacakkappavattana by teaching the doctrine of the Middle Way: i.e. he did that because that specific audience (who had met him before enlightenment) thought he must be wrong (and therefore, perhaps, unenlightened and not worth listening to) because he had stopped performing austerities:

Friends, here comes Gotama the contemplative: living luxuriously, straying from his exertion, backsliding into abundance.

So they for example could be called a "category-of-people-who-think-you-need-to-perform-austerities-in-order-to-find-enlightenment", or the "category-of-people-who-thought-they-could-address-the-Tathagata-by-name-and-as-'friend'".

So, lots of categories.

And as I said perhaps it's not sane to try to categorize (so the preceding was an example of wrongly categorizing).

Maybe there are as many different categories are there are different ignorances/delusions. :-)

It's not clear which Buddhist school you're asking about but the question is tagged dharmas. Apparently, a translation of one of the versions of the Bodhisattva vows is,

Sentient beings are numberless; I vow to save them all.
Desires are inexhaustible; I vow to end them all.
The Dharma Gates are infinite; I vow to enter them all.
The Buddha Way is unexcelled; I vow to attain it completely.

Another translation of that is, "the dharmas are infinite".

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If your question was asking how to teach (asking how you could teach) Buddhism to different (categories of) people, there were some answers to that here: How to explain what Buddhism is?

The question, though, might be like asking, "My mum has given people little presents which meet their particular needs. What are the different categories of people?" Well, there's a category of people who'd appreciate being given a new, easy-to-use, pink umbrella for their own use; there's a category of people who'd feel touched, by a hand-sewn sachet with dried lavender and their name embroidered on it; etc. If it really is 'tailored' and enlightened it might defy categorization, automation, generalization.

In case it matters, when you asked the question my first thought/memory was that there was only one category: i.e. "beings with little dust in their eyes (SN 6.1)". Or depending on how you interpret that sutta, a different (single) category i.e. "those who would hear" (sometimes translated, "those with ears"). Or a more inclusive set of categorizations,

Surveying the world with the Buddha-eye, the Blessed One saw beings with little dust in their eyes and beings with much dust in their eyes, the keen and the dull, the good and the evil, those easy to teach and those hard to teach, and some who live seeing blame (for wrongdoing) and fear (of pain) and in the hereafter.

Just as in a lotus pond of the blue or white or red lotuses, some lotuses might be born in the water, grow up in the water, and thrive while submerged in the water, without rising out of the water; some lotuses might be born in the water, grow up in the water, and stand up at an even level with the water; some lotuses might be born in the water and grow up in the water, but would rise up from the water and stand up in the water without being soiled by the water––

So, too, surveying the world with the Buddha Eye, the Blessed One sees beings with little dust in their eyes and with much dust in their eyes, with keen faculties and with dull faculties, with good qualities and with bad qualities, easy to teach and difficult to teach, and a few who dwelt seeing blame and fear in the next world.