Where in the canon does the Buddha teach about mindfulness?

• Abhidhamma

Therein what is mindfulness-awakening-factor? Herein a monk is mindful, furnished with excellent mindfulness-penetration, he remembers, remembers constantly, what has long been done and long been said (concerning release). This is called mindfulness-awakening-factor. (1)

That which is mindfulness, recollection, recall, mindfulness, remembrance, bearing (in mind), not losing, not confusing, mindfulness, the Faculty of Mindfulness, the Strength of Mindfulness, Right Mindfulness: this is called ‘mindfulness.’

The mindfulness which on that occasion is recollecting, calling back to mind; the mindfulness which is remembering, bearing in mind, the opposite of superficiality and of obliviousness; mindfulness as faculty, mindfulness as power, right mindfulness—this is the faculty of mindfulness that there then is.

• Sutta

Suppose, monk, that there were a royal frontier fortress with strong walls & ramparts and six gates. In it would be a wise, experienced, intelligent gatekeeper to keep out those he didn't know and to let in those he did. A swift pair of messengers, coming from the east, would say to the gatekeeper, 'Where, my good man, is the commander of this fortress?' He would say, 'There he is, sirs, sitting in the central square.' The swift pair of messengers, delivering their accurate report to the commander of the fortress, would then go back by the route by which they had come. Then a swift pair of messengers, coming from the west... the north... the south, would say to the gatekeeper, 'Where, my good man, is the commander of this fortress?' He would say, 'There he is, sirs, sitting in the central square.' The swift pair of messengers, delivering their accurate report to the commander of the fortress, would then go back by the route by which they had come.

"I have given you this simile, monk, to convey a message. The message is this: The fortress stands for this body — composed of four elements, born of mother & father, nourished with rice & barley gruel, subject to constant rubbing & abrasion, to breaking & falling apart. The six gates stand for the six internal sense media. The gatekeeper stands for mindfulness. The swift pair of messengers stands for tranquillity (samatha) and insight (vipassana). The commander of the fortress stands for consciousness. The central square stands for the four great elements: the earth-property, the liquid-property, the fire-property, & the wind-property. The accurate report stands for Unbinding (nibbana). The route by which they had come stands for the noble eightfold path: right view, right resolve, right speech, right action, right livelihood, right effort, right mindfulness, right concentration."

Here is a good description of mindfulness (sati) directed to the body. It's similar for mindfulness directed to other things.

“Bhikkhus, suppose that on hearing, ‘The most beautiful girl of the land! The most beautiful girl of the land!’ a great crowd of people would assemble. Now that most beautiful girl of the land would dance exquisitely and sing exquisitely. On hearing, ‘The most beautiful girl of the land is dancing! The most beautiful girl of the land is singing!’ an even larger crowd of people would assemble. Then a man would come along, wishing to live, not wishing to die, wishing for happiness, averse to suffering. Someone would say to him: ‘Good man, you must carry around this bowl of oil filled to the brim between the crowd and the most beautiful girl of the land. A man with a drawn sword will be following right behind you, and wherever you spill even a little of it, right there he will fell your head.’

“What do you think, bhikkhus, would that man stop attending to that bowl of oil and out of negligence turn his attention outwards?”

“No, venerable sir.”

“I have made up this simile, bhikkhus, in order to convey a meaning. This here is the meaning: ‘The bowl of oil filled to the brim’: this is a designation for mindfulness directed to the body. Therefore, bhikkhus, you should train yourselves thus: ‘We will develop and cultivate mindfulness directed to the body, make it our vehicle, make it our basis, stabilize it, exercise ourselves in it, and fully perfect it.’ Thus, bhikkhus, should you train yourselves.”
SN 47.20

There are 56 samyutta's in the SN, the 47'th SN is devoted to the topic of "mindfulness" (sati).

SN 47 Sati-'paṭṭhāna 🐘 Saṃyutta https://lucid24.org/sn/sn47/index.html (disclosure: my website, my translations are derived from B. Sujato)

The first 10 suttas especially are particularly important on the subject. But even reading various English translations of those 10 suttas, you can get quite a different idea of what the practice of sati means. With a bad translation it's easy to arrive at wrong interpretations of 'sati' that totally miss the mark.

So SN 47 should be your first stop for authoritative passages, other suttas are mentioned here, along with brief explanations: https://lucid24.org/sted/8aam/7sati/index.html

A great way to check if your understanding of 'sati' is correct, is with the fortress sutta simile, of AN 7.67. https://lucid24.org/an/an07/an07-0067/index.html

If it's a correct sati, then the wise general who guards the gate is going to be effective. If it's a wrong understanding of sati, like many modern "mindfulness" teachers teach, then the guard is going to let bad guys waltz right into the fortress.