The dimensions you are refering to, sound about like a standard bathroom vanity size. That would be the cabinet that the countertop sits on top of. A self rimming sink then drops into a hole cut into the countertop.
Another option would be a single unit that is both the countertop and sink in one. You can buy what is called a cultured marble sink/vanity top that is one piece sized for your particular vanity. Also available in solid surface materials.
If you are replacing just the sink and your counter top is in good shape, you can remove the sink from the countertop and replace it with a new sink that is the same size and shape. May have to go to plumbing supply place rather than the big box stores.
If you're going to replace the counter top, then find the type of top you like as well as the sink you like. Put the template down on the counter top and cut the hole for the sink.
Before you cut the hole, or decide on a sink, make sure the sink will fit the space in the vanity cabinet. If you have drawers on the top portion of the cabinet, the sink needs to fit between them. Most "bathroom" sinks are going to be fine, but better to check beforehand.