How did I know? In addition to the stem and leaf shape, the photo included the plant tag - setting up a 'home run'. This told me it was a tuberous begonia Begonia tuberhybrida. Tuberous plants form tubers - think potatoes. It makes sense, really, that the presence of a tuber would allow any plant to go dormant and later use the sugars and starches stored in the tuber to re-grow.
Since this begonia is in a pot, it just needs to be kept in a cool dry place in the house - not in a dark closet or anywhere that you'll totally forget about it. While it doesn't need water during the winter, you want it to be where you'll notice when the new buds form on top of the tuber. This will be your signal to start watering again. If your potting soil has seen better days or you want a bigger root system (for a bigger plant), feel free to dig the tuber up (around February) and repot it (at the same depth that it was) with some fresh potting mix. Keep the soil moist but avoid watering directly on top of the tuber as you don't want it to rot.
Once your begonia begins growing again, use a dilute (half the label rate) solution of balanced fertilizer every couple of weeks for best results. (I have to admit, I've not fertilized anywhere near that often and always have beautiful plants - just avoid over-fertilizing at any one time - not an issue at my place.) Once you've passed the frost-free date in your area, move your begonias to a shady location outside and prepare for the blooms. Peak seasons are spring and fall, so don't worry if your plants bloom less in the heat of summer.
Tuberous begonia entering dormancy.
Tuberous begonia in bloom.
