Winter hiking in Yosemite National Park
The Tioga Road, which reaches 10,000 feet (3,000 meters) in elevation, is closed altogether in the winter, and the Glacier Point Road, with an average elevation of around 7,000 feet (2,100 meters), is closed to automobiles fives miles in (at Badger Pass, Yosemite's alpine ski area) and transformed into a cross-country skiing and snowshoeing trail. Yosemite Valley at an altitude of 4,000 feet (1,220 meters), is open year round and fairly temperate by High Sierra standards. It gets enough snow to spawn the occasional Calvinesque snowman, but not so much that you would practically use snowshoes or skis (or not often and not for long, at any rate).
The Sierra snow gods typically don't get really busy until January; chances are about 50-50 that you'll have enough precipitation for snow activities in December, and it's not unheard of for you to have to postpone your snow jones until late January.
Most of the trails in Yosemite Valley that don't lead up the valley walls are open year round. Bridalveil Fall, Lower Yosemite Falls, Mirror Lake, and the Vernal-Fall-footbridge section of the Mist Trail are all hikable, except after unusually heavy snowfalls. Artist Point, too, is accessible nearly all the time, and the valley loop trail, which winds from one end of the valley to the other, is always open.
Trails going from the valley floor to the rim are harder to predict. The Snow Creek Trail stays open year round. The bottom half of the Upper Yosemite Falls Trail also stays open, but a gate just past Om My Gosh Point near where the trail rounds the corner to a view of the middle cascades, can be closed during wintry conditions, and even when it's open, there's a good chance of snow on the upper half of the trail. The 4-Mile Trail usually stays closed for the winter. Though the Mist Trail closes past the footbridge, you can divert there to the John Muir trail and get access to the tops of Vernal and Nevada Falls via the Muir Trail's more roundabout route. Sometimes the very end of the trail that leads to Nevada Fall will be closed due to ice, though, and the Muir Trail can have difficult snowy stretches even when it's open. The Pohono Trail from the Tunnel View up to Stanford and Crocker Points on the valley rim doesn't usually close, but snowy conditions can make the going difficult.
The Hetch Hetchy region is lower than Yosemite Valley, and hikes there, such as Wapama Falls are usually accessible in the winter. Given that it lies below 4,000 feet (1,220 meters) elevation, the Hetch Hetchy area is probably preferable in the winter when it's not so hot.
The trailheads for two of Yosemite's sequoia groves (Merced and Tuolumne) are reachable by car year round, but they're at much higher altitude than Yosemite Valley, and you may need snowshoes, depending on how wet of a winter it's been. The road to the Mariposa Grove closes for the winter, which means that you'll have a two-mile hike to get to the trailhead, and you'll likely need snowshoes in the grove itself.
In southern Yosemite, the Wawona Meadow Loop and Wawona Swinging Bridge trails are accessible year round. The Chilnualna Falls Trail generally doesn't close, either, but could have snow on it.
The Yosemite webcams can also give you an idea of how much snow is lying about the place and whether or not there's enough water flowing over Yosemite Falls to tempt you to try the trail.