In the five paramitas, what is patience and why is it on the list?

"Patience" can be seen as passive, or active. If I simply endure whatever occurs (even, perhaps, if I am aware of it, and of my experience of it), I am patient in a passive sense; but to "suffer" can also have more active senses.

Perhaps the added element would then be intentionality, or an aligning of my will with, the fact of my experience. At which point, the will can evaporate out of the interaction, leaving reality.

Acceptance, to put it more simply. A whole-hearted engagement with my experience of suffering (or of whatever I am experiencing, including "good" experiences), unconditioned by judgement, reservations, expectations - by "thinking", as it works against reality - and perhaps grounded on the assurance that this, and nothing else, is the reality of the universe for me, now, and to imagine / crave / hope for / plan for / long for anything other than what my experience is, here and now, is a form of self-annihilation.

To add few points to nemo's answer which is basically correct.

As per Chogyam Trungpa Rinpoche, the Buddhist definition of patience (ksanti) is freedom from aversion (dosa). Such aversion often manifests as one of: aggression, irritation, anger, resentment or depression.

An important idea to get is that fully developed transcendental patience (kshanti-paramita) is not based on suppression! It is not forbearing an annoying person or cold winter or the sounds distracting one from the sitting meditation while silently hating it!

Instead, transcendental patience is based on A) realizing the nature of the problem, B) realizing the nature of one's own aversion and C) realizing the constructive attitude leading towards the Higher Goal.

• Realizing the nature of the problem is to see and acknowledge the objective causes of the situation. It may be atmospheric conditions, or other person's pain and confusion due to their own childhood living conditions or whatever. By analyzing and tracing back the sources of the issue you no longer see it at the superficial level, which removes a lot of negativity.
• Realizing the nature of one's own aversion is to see and acknowledge own's own preconceptions / stereotypes / attachments that contribute to superficial vision of the situation.
• Realizing the constructive attitude is to abandon the reactive response to the unacceptable situation, a response designed to protect one's ego and to uphold one's preconceptions - and instead to chose a course of action that optimizes the future outcomes. If conflict with a person is brewing, would passive-aggressive behavior and patronizing improve the situation, or should you change the strategy and try and make the person your ally - or on the contrary to come out openly with your issue. Sometimes (usually when you are in the role of a parent or teacher or manager) manifesting aggression is more constructive than suppressing the resentment. What's important to keep in mind is the Higher Goal. Instead of getting obsessed with an obstacle causing our irritation, we can almost always find an alternative way to work around it and get back on the path to the goal.

The point here is that transcendental patience is not a patience of a victim, it is patience of a master.

Development of patience-paramita is related to Buddhist training point known as "taking all bad circumstances onto the path". Instead of mentally labeling problems as punishment or aggression coming from the world, we mentally thank the world for the opportunity to improve our practice.

A very specific part of this is the practice of taking all the blame upon oneself. When we meet any unacceptable situation with a very bold and ego-less step of taking all the blame, taking responsibility unconditionally, regardless of who is right -- something magical begins to happen. The bubble of ego energy that was getting pumped by the aversion bursts and a true opening can take place, a true awakening and opening of the eyes. This opening is the paramita of patience.

Patience is a mind that abides in its natural state unaffected by harm and suffering. It strongly abides in the Dharma.

The division is threefold:

1. Patience bearing suffering
2. Patience not paying heed to those who cause harm
3. Patience stable in the Dharma

To be a fully qualified perfection, it must be conjoined with the wisdom realizing emptiness and effortless bodhicitta. The perfection of patience is so perfected on the level of the 3rd ground (bhumi) on the supra-mundane path.

The six perfections are related to the two accumulations of (1) merits and (2) wisdom that lead to buddhahood. The accumulation of merits is related to the first three perfections: generosity, ethics and patience. (that partly explains why patience is a perfection)

Indeed, from the point of view of the result: the fruit of patience [and so forth] is buddhahood. The two essential causes for that fruit are: (1) myself practicing patience, and (2) the enemy that is the object of patience. One can not achieve enlightenment in isolation, since it depends on others. As in Shantideva:

6.108 Because I am able to practice this, He is worthy of being the very first to be given The fruit of my patience, For in this way he is the cause of it.

The relationship between ethics (2nd perfection) and patience (3rd perfection) is such that if one wants to practice the three aspects of ethics, it will involve hardships. This is where patience comes into the picture. We need to learn how to endure those hardships. (that also explains that it is a perfection, since it is in that sequence)

In terms of benefits, as anger is greatly disadvantageous since it destroys roots of virtue (the meaning of which is discussed at length by Tsongkhapa in the Middle Length Lam Rim and the Chen Mo as well), patience, on the other hand, is the direct antidote to anger. Thus, it prevents one from getting his roots of virtue destroyed: As in Gyaltsab Je's commentary to Shantideva:

Although one meditates the virtue of generosity and the like for eons, they are destroyed by the fire tongue of anger. Therefore, one needs to generate the force of patience again and again and not give anger any chance.

Patience has other benefits as well, such as:

1. Being a way to purify negativities
2. Being “the best way to accumulate merit” (I quote here one of my teachers, Kyabje Geshe Gyaltsen)
3. Increasing our qualities and happiness (i.e. if one did not react with anger but practiced patience the mind would not be affected by the harm and pain)
4. Being a way to take the essence of this precious human rebirth

That cultivating patience is a way to accumulate merit is shown in Shantideva:

6.103 If by my own fault I am not patient with this, Then it is only I myself hindering Involvement in the cause of merit.