What does sati mean?

The way it's used in Buddhism, sati/smrti means continuously remembering something, being mindful of something, continuously keeping something in mind. The traditional metaphor is the mindfulness of hunger -- when you are very hungry you don't forget that you are hungry.

If I remember correctly, before Buddhism, in Hinduism (or whatever that conglomerate of teachings and rituals can be called) smrti used to refer to rote learning. So basically, you repeat some Vedas text until you memorize it by heart.

Here we have two components: keeping something in mind, and repeating something until it sinks down. This is basically what smrti is. Specifically, kayagata-smrti -- mindfulness rooted in the body -- is continuous awareness of one's bodily sensations, first and foremost somatic components of emotions. ana-apana-smrti -- mindfulness of breathing in and out -- is continuous awareness of the breathing, again mostly because breathing is always the picture of one's state of mind. Both of these can be both meditation as well as post-meditation activities -- and it is the post-meditation that is referred to as smrti, because the challenge is not to forget it, as you are walking about your business.

Again, the important thing to remember that smrti is never just mindfulness (as in, open-ended awareness) -- it is always mindfulness of something. It does matter what we are mindful of, because whatever it is, the more you feed it the stronger it gets.

Usually in Buddhism we train to remember whatever our teacher deems useful for us to remember at this particular stage. It could be the lower abdomen, could be the diaphragm, could be the face, could be the hands, could be the feet, could be the head and shoulders, could be Buddha-Nature, could be Emptiness etc.

More broadly, at "hinayana" stage the training is focused on very low level immediate experiences: so what you are "remembering" (paying attention to) is the whole phenomenological setup -- bodily sensations (heartbeat, breathing, face muscles, posture), emotions (somatic projections aka chakras), and the state of mind (excitement level, scattering level, dimness level, viscosity level etc) -- as a whole. So basically we are learning to be more self-aware, to have a better insight into our internal condition -- via all kinds of hints and symptoms. At this level it's not just being aware though, it's fixing whatever looks unskillful/unhealthy. Noticed the heartbeat increasing -- stopped arguing and switched the strategy of persuasion. This kind of stuff.

Even more broadly, smrti means remembering Dharma at large, as the all encompassing context of our trainee's life, as opposed to letting one's inner frame of reference be force-switched by the circumstances. This mastery over one's context is developed through five levels (panca bala): faith, effort, mindfulness, concentration, and wisdom.

1. At entry level, mindfulness starts as faith or inspiration. Inspired by an encounter with Sat-Dharma, our mind gets the bug of Buddha-fever. Unable to resist the urge to find out more, we keep thinking about mysterious Dharma.
2. At beginner's stage mindfulness means applying effort to constantly check one's current physical, verbal, and mental activity against the dharmic guidelines. Am I being driven by an afflicting emotion? Is my mind presently biased by an attachment? Is my current action motivated by an egoistic intent?
3. The next level is when staying always mindful of the context of Dharma becomes automatic and no longer requires effort, which means one is now always operating in the phenomenological context, as I described two paragraphs ago.
4. At middling level mindfulness grows into concentration (samadhi) which means we can now design, assemble, and maintain any context, any frame of reference, any mood we choose. If we want to feel happy -- we can contrive the context in which we will be happy. If we want to feel motivated -- we can contrive the frame of reference that will make us passionate. And so on. This is also known as generation-stage meditation. The role of smrti here is to keep the contrived context on.
5. Finally, at advanced level mindfulness becomes wisdom (prajna), which means seeing all contexts at the same time and freely juggling their elements as required by a situation at hand. This is like being a lawyer, in the sense that you see every situation from all possible angles at once, can speak from the perspective of a certain frame of reference, depending on whom is it that you're addressing, and even speak in multiple frames of reference at the same time. Why this is still smrti is because it involves staying mindful of all those different contexts.