Stalking is the act of following someone or something very closely and watching its every move. If you're a very famous movie star, it's quite likely that someone is stalking you even as you read this sentence!
The verb to stalk means to pursue carefully, and often stealthily. It was originally used to describe hunters following their prey and waiting for the precise moment to attack. In its more recent, colloquial use, stalk and its gerund form stalking, refer to the act of pursuing a person, often because you have an unhealthy obsession with them. Stalking your ex-girlfriend to see if she’s dating someone may seem like a good idea, but it’s unwise and illegal!