This is an excellent question, and one that isn't asked often enough. Presumably, this is another one of those art definitions we are all supposed to know - most likely by osmosis, or some other telepathic means, because (heaven forbid) you wouldn't want to ask a "stupid" question at some Art World function. (Well, you might, but I wouldn't. At least, not ever again.)
Anyway, the answer is divinely simple. Contemporary just means "art that has been and continues to be created during our lifetimes". In other words, contemporary to us.
Now, of course, if you are 96-years old and reading this (By the way, congratulations, if this describes you. Way to keep up with the times!), you can expect a certain amount of overlapping between "Contemporary" and "Modern" art in your lifetime. A good rule of thumb is:
- Modern Art: Art from the Impressionists (say, around 1880) up until the 1960's or 70's.
- Contemporary Art: Art from the 1960's or 70's up until this very minute.
Here at About Art History, 1970 is the cut-off point for two reasons. First, because it was around 1970 that the terms "Postmodern" and "Postmodernism" popped up - meaning, we must assume, that the Art World had had its fill of Modern Art starting right then.
Secondly, 1970 seems to be the last bastion of easily classified artistic movements. If you look at the outline of Modern Art, and compare it to the outline of Contemporary Art, you'll quickly notice that there are far more entries on the former page. This, in spite of the fact that Contemporary Art enjoys far more working artists making far more art. (It may be that Contemporary artists are mostly working in "movements" that cannot be classified, due to there being around ten artists in any given "movement", none of which have shot off an email saying that there's a new "movement" and "could you please tell others?")
On a more serious note, while it may be hard to classify emergent movements, Contemporary art - collectively - is much more socially conscious than any previous era has been. A whole lot of art from the last 30 years has been connected with one issue or another: feminism, multiculturalism, globalization, bio-engineering and AIDS awareness all come readily to mind as subject matter.
So, there you have it. Contemporary art runs from (roughly) 1970 until now. We won't have to worry about shifting an arbitrary point on the art timeline for another decade, at least. Go, be of good cheer, and fear not the term "Contemporary Art".
See also: What is "Modern" Art?