I have a Aster DEM, and from that i get the drainage network, and now i need to calculate the drainage area, but i don't know if the drainage area is equal to the basin area.
If I've interpreted your question correctly, it sounds to me like you are interested in deriving a network of stream lines (drainage network) from a DEM. If this is the case, then the drainage area that you are referring to is actually a raster of upslope contributing area (also known as the specific catchment area or flow accumulation). The flow accumulation grid is essentially a raster that shows for each pixel the number of pixels (or area) that lie upslope of the pixel and that are connected to it through a flow path. The value of flow accumulation is a monotonic step function increasing downslope such that river valley bottoms tend to have considerably higher values than their adjacent hillslopes. This is the reason these rasters are used for automated DEM based stream mapping. A drainage basin, or in some places what is more commonly known as a watershed, is the area that drains to a single point, usually an outlet point along a stream network. They are related concepts and easy to confuse, particularly given how many regional synonyms each term has.
Don't worry about the downvotes. Sometimes people forget that GIS is applied to the study of biophysical phenomena just as often as it is applied to social phenomena. It's a very diverse field!