Mindfulness, to put it as plainly as possible, is the ability to observe one's own internal sensations with the calm clarity of an external witness. It has been described by the great Buddhist monk Nyanaponika as "the clear and single-minded awareness of what actually happens to us and in us, at the successive moments of perception."
I came across the concept and practice of mindfulness when I first started trying to actively manage my Obsessive-compulsive disorder (OCD) about 3 years ago. In his book Brain Lock Jeffrey Schwarz advocates using mindfulness as part of his 4 step approach to managing the condition, basically through being mindfully aware of distressing thoughts as they arise and working around them rather than trying to argue with them or push them away (which - owing to the mechanism of OCD - makes them more likely to recur).
As well as being a useful technique for managing a troublesome psychiatric disorder, it turns out mindfulness is a central tenet of buddhism. By paying non judgemental bare attention to one's internal states one is able to observe things as they really are rather than continually reacting to thoughts, feelings and emotions. Two quotes from the Buddha:
"Mindfulness, I declare, is all-helpful"
"All things can be mastered by mindfulness"
Mindfulness can be cultivated by regularly meditating - most commonly by focusing on observing one's breathing. There are quite a few guided meditation mp3s out there on the internet - the introductions and lead throughs by Kamalashila linked to here are pretty good, I think. I try and do a 15 or 20 minute meditation on most days and certainly find it beneficial.
"May all beings be happy" :-)