Nope:-) It's not exactly easy to define "syzygy" in a few words (I only do it in advanced grad courses, and trust me that it takes me a few weeks of work!:-), so this is the wikipedia link...
Basically (but very basically!), syzigies are the relations (or the relations among the relations, and so on, until you stop thanks of a theorem of Hilbert...) among polynomials....
Another cute property is that "syzygy" is, or so they say, the only word in Canadian or British English (but not American!) that when you write it by hand, it ends up under the "line" 6 times in a row... ;-)