The roads that are open are kept pretty clear of snow most of the time. It's hardest during a snowstorm, but they don't take long to clear them.. The biggest problem is when it's warm during the day and the roads ice up at night and stay icy in the morning and on shaded curves.
The roads that are snowbound are closed until late spring (late May or June usually). That is, Glacier Point Road and Tioga Road (Highway 120 East over the Sierra Nevada crest). The road to Wawona Grove is also closed,
Here's a good webpage to check road conditions. Calling the recorded number is better during a storm as they seem to keep it more up-to-date for some reason.
http://www.nps.gov/yose/planyourvisit/conditions.htm
The Yosemite Valley webcams also give a rough idea of what it's like:
http://www.yosemite.org/vryos/