How do I replace a bathroom sink without having to retile the counter?

Get a sink that fits in the hole on your vanity and you don't have to replace the vanity or the counter top.

A photo would help. Is the sink a separate unit or part of the countertop?

Hi Pje: You can usually just swap out the sink. Take measurements to the home improvement store and get one that fits and that you like, checking that the holes for the faucets match, unless you are going to get new faucets. Take a couple of pictures of your sink to show the people at the Help couter. The screws to remove the old sink are under the counter and there might be some little flaps that hold in the sink. Ask at the help counter as to how to proceed and get some of that plumbers putty or whatever you need to keep it steady. Don't forget to turn off the water:) You can do this, I did, and I'm not all that handy:) Good luck