The idea is based on the fact that silicon has valence number of four. This means that silicon atoms can be arranged in rings and in long chains that may be useful to create structures upon which biological molecules could be built. However, there are many drawbacks that must be overcome for life to efficiently use silicon as its basis.
Silicon based lifeforms would not have organic molecules used within Terran life. DNA would not be the basis for such life. However, the silicon based molecules may not be stable without an added level of complexity because silicon has a larger atomic radius and mass than carbon. It also has more difficulty forming stable molecules, particularly where water is present.
Even still, it is a possibility. For some reason, many Sci-Fi depictions show silicon based life as being rocklike in appearance. I'm not sure where this idea comes from. It's a bit like assuming carbon based lifeforms look like a lump of coal. I have a feeling that if we do discover silicon based lifeforms, they may resemble us more than many expect.
Reference: Wikipedia article and “Are Aliens Among Us?” by Paul Davies, Scientific American December 2007