I bought this plant from a home repair store 3 days back, but I don't know what it is. Now it seems that its decaying. What is this plant and what do I need to do to take care of it?
Begonias commonly get gray mold (Botrytis). Since this fungus moves from dead tissue into living, pick off any dead flowers or leaves as soon as you can. Try to keep the foliage and flowers dry.
Does your pot have drain holes? The begonia is very sensitive to wet feet. Allow the plant to dry out somewhat in-between watering. In most areas this plant is an annual. I'm not sure where you live, if you can and if you have a covered patio, this plant would thrive outside.
Cut flowers off as soon as you can. Annuals have one purpose in life and that is to produce seed. Thus lots of flower display. But once it's made seed the whole plant just starts relaxing. It has done its job. To help the plant remain vigorous and produce flowers all season, cut the flowers off as soon as possible.
All that energy going into making flowers (seed) is diverted back to the plant. If it hasn't already been fertilized by a greenhouse or the nursery (something else you should ask if you go back to the store is when was it fertilized last, what kind of fertilizer) fertilize it with a low-nitrogen fertilizer. If you use a higher percentage of Nitrogen in relation to phosphorus and potassium you will get lots of vegetative growth and little or no blooms.
Put it in indirect sunlight when indoors, but still a brightly lit area. For the summer it would love to go outside on the covered porch but protect from wind. If you have winter, bring it back inside by gradually increasing the time inside. It'll need to get used to the dry air again. And keep an eye out for spider mite or other insects. Gray mold, too as Eric brought up!