If your weld isn't on a piece of metal that will show, or if you don't care about how the weld looks, then you are done with your weld. However, if the weld is showing or you are welding something that you want to look nice then you will most likely want to grind down your weld and smooth it out.
Slap a grinding wheel onto an angle grinder and get started grinding on the weld. The neater your weld was the less grinding you will have to do, and after you have spent a whole day grinding, you will see why it's worth it to keep your welds neat in the first place. If you use a ton of wire and made a mess of things it's ok, it just means that you might be grinding for a while. If you had a neat simple weld though, then it shouldn't take too long to clean things up.
Be careful as you approach the surface of the original stock. You don't want to grind through your nice new weld or gouge out a piece of the metal. Move the angle grinder around like you would a sander so as not to heat up, or grind away any one spot of the metal too much. If you see the metal get a blue tinge to it you are either pushing too hard with the grinder or not moving the grinding wheel around enough. This is can happen especially easily while grinding thing sheets of metal.
Grinding welds can take a while to do depending on how much you have welded and can be a tedious process - take breaks while grinding and stay hydrated. (Grinding rooms in shops or studios tend to heat up, especially if you are wearing leathers). Wear a full face mask when grinding, a mask or respirator, and ear protection. Make sure that all your clothing is neatly tucked in and that you don't have anything hanging down from your body that could get caught in the grinder - it spins fast and it can suck you in!
When you are done your piece of metal might look something like the one in the second photo pictured below. (Or maybe better as this was done by a few Instructables Interns at the beginning of the summer during their first welding experience.)