1. When faced with this issue, it is best to ignore and move on. Do not show any interest with this person for he or she will feed on your reactions.
2. Do not react on what this person does to you. Don't speak or show any sign that he or she is really getting to you, because this will keep them going.
3. Most of the time you dont know the person that you are talking to. You need to stay on your toes because it is very easy to trick anyone online. Don't put any of your life history and personal information including your full real name online. I know you would think this is a given but people do it all the time.
There are quite a few places a stalker can find you. Myspace is a perfect example. They did an investigation and found thousands of pedophiles on this site. Young girls are the easiest target but so are desperate people. If you are freshly out of a relationship, you could be the perfect target. Watch yourself. If the person does not stop after being asked, you might want to consider legal action. It's happened to me. Not the legal action part but being cyberstalked. I no longer use my Instant Messenger and I feel much safer.