The Pali word is Sati, which derives from the verb Sarati, which literally means to remember. If you look at how the word is actually used in the texts, there seems to be three distinct senses of the term.
The first sense is the most general, and it just means memory, or the ability to recall events and information. This isn't a major meaning of the term Sati in the Buddhist texts, but it does show up occasionally.
In a second sense, Sati has a more specific meaning of focusing the mind on an object. This is more commonly used in the form of a suffix, so for example
Ānāpānasati means setting the mind on the Ānā and pāna, the in-breath and the out-breath.
In the last sense, it has a much more specific meaning. It means the setting of the mind on an object of ultimate reality, meaning setting the mind on direct experience itself, which can be divided up into four categories of body,feeling, mind, and dhammas, meaning physical sensation, the experience of something as pleasant, unpleasant, or neither, awareness, and various mental qualities that occur in the mind such as emotions.
This list is called the four Satipatthana, and the correct setting of the mind on them is called Samma-Sati, or right mindfulness, which is the seventh factor of the Noble Eightfold Path. The standard definition found in many places in the Suttas is as follows:
And what is right mindfulness? There is the case where a monk remains
focused on the body in & of itself — ardent, alert, & mindful —
putting aside greed & distress with reference to the world. He remains
focused on feelings in & of themselves... the mind in & of itself...
mental qualities in & of themselves — ardent, alert, & mindful —
putting aside greed & distress with reference to the world. This is
called right mindfulness.
http://www.accesstoinsight.org/tipitaka/mn/mn.141.than.html
There are other aspects of the definition which the Ven. Yuttadhammo has already mentioned in his answer as well