Begonias are a huge family of plants ranging from indoor to outdoor plants. When we think about begonias the most common outdoor annual types are the fibrous begonias and tuberous begonias.
Fibrous begonias are so named because of their root system. They have a fine root system that spreads into the soil but isn't used to store energy. These are what I would call the true annual begonias, though they could technically be brought indoors as a house plant. With these plants, it's easiest just to buy them as plants in the spring and plant them out in your garden. When the frost comes in the fall, just let them die with cold and start again next spring.
Tuberous begonias are a little different. They get their name from the large tuber they form underground to store energy for a dormant period. They can't take frost so you can't leave them in the ground, but you can keep them year after year. You can sometimes purchase these plants in the spring with other begonias, though they will have rather small tubers by the end of the year. If you really want to get nice begonias, though, you can purchase the tubers in the spring. Ideally, you want to plant them as early as you can since they can take quite a while to flower. Typically, a tuber is at least a year old since it takes that long for the tuber to form. Larger and older tubers will produce larger and healthier plants with more blooms. Some of the best Begonia tubers come from Belgium which can cause problems for us here in Canada since the cold can create problems for shipping the tubers.
When you do look at a begonia tuber, the first question that pops to mind is "Which was is up?" Unlike most other bulbs where it's pointy side up, there is no point on a begonia bulb. Instead you will find the bulb looks like a flattened ball with a large dimple on one side and sometimes a little fur on the domed side. The domed side is the root side and the indented side is the plant side or "this side up." Plant the bulb so it's slightly under the surface of the soil. Begonia tuber growth is triggered by heat more than anything so try and keep the the soil nice and warm and moist. You can either plant them in a smaller container, at least twice the size of the bulb, and then transplant them outdoors later. Alternatively, and my personal preference, is planting them in a large decorative planter which can just be moved outside in the spring. Having the bulbs in a planter also gives you a little more flexibility when the fall comes since you're less pressured to dig up the bulb and instead can take the whole planter in before it gets too cold.
When the fall comes, the bulbs need to be brought indoors before it freezes. Simply cut back all the leafy growth, then dig up the bulb. Remove most of the roots so you end up with just a bulb (tuber technically). Leave the bulb in a cool dry place for a little while to dry a bit which prevents fungus or rot. Once they are a somewhat dry, after a week or two, place them in a cardboard or other breathable container filled with dry peat moss or sawdust. Place them in a cool, dry, and dark place and leave them until the spring. If it's too light or too warm, the bulbs will start to grow too early in the winter.
In the winter, it's good to start your bulbs early to get the most growth you can. February is a good time to start them if you can which gives the begonias a good head start for the spring. It's quite easy to grow begonias and it really isn't that hard to keep them year after year. Watch your bulbs grow larger and larger each year and enjoy the massive display of colours in your shady garden.