I believe you have a Rex Begonia (Begonia Rex-cultorum).

As to the care — Rex begonias prefer a drier soil. Keep the soil to the dry side but not to the point of wilting. When watering, you should water in a way that the foliage does not get wet. Leaf moisture can cause bacterial leaf spot. Lift the leaves and pour the water directly on the soil.

When you re-pot the begonia into the decorative pot use a peat-lite soil media. Heavy soils are detrimental to the Rex begonia. I would recommend Baccto lite soil mixture. It is a peat based light soil medium that you can find at your local nursery & garden center. Also, make sure that the plant stay at the same level when you pot it. This means that the soil you see at the top right now should still be seen when you re-pot it.

You will need to place this plant in an area with bright indirect light and a fairly cool atmosphere. Most house temps are sufficient for the Rex begonia.

Rex begonias are moderate feeders and a slow release granular fertilize is optimum. You can use a liquid fertilize if easier to apply. Occasionally apply a small amount of fish emulsion – it contains some micro nutrients that are beneficial.

Blooming occurs during short-day periods like winter.

Things to watch for with the Rex begonia are bacterial leaf-spot and root rot fungi. Both are controlled with your watering practices. Keep water off foliage to prevent bacterial leaf spot and don't over water to prevent root rot.