Is a new term for me, but not a new concept. It has its roots in buddhism, but it's become a popular way to handle emotions, outside of any spiritual connotations.
When I recently read about mindfulness, I had a "So that's what it's called! Other people do that too!" moment.
Mindfulness is an attempt to view the present, instead of focusing on the past or the future, as we are used to doing. The moment-by-moment awareness of what we're doing, what's going on around us, and what's going on in our head is mindfulness. When you practice being mindful, it's like you're seeing everything clearer, and like you're seeing everything at once. You see your circumstances and emotions much more clearly, in a detached way, as if you're someone else looking in.
You notice all sorts of things that might be filtered out of your perception normally. You notice the beauty of obscure things.
I'm not an expert on initiating mindfulness, for me it's like flipping a switch. I have to make the effort to be mindful (and it is an effort, to begin and maintain for any period of time), but then I enter that peaceful, observant, quiet space, at least for a few moments of my day.
P.S. I'm pretty sure artists and kids are better at entering mindfulness than normal people - that's why they notice so much that other people don't see.