Monitoring involves paying close attention. It's a type of systematic observation, like the monitoring of criminals by the police. Kids who are up to something don’t like their parents’ monitoring their every move.
To monitor is to keep an eye on someone or something, often through recording devices. Monitoring, then, is the act of keeping tabs on others. A lot of monitoring is done by law-enforcement agencies. The monitoring of potential terrorists is done by the CIA, and the monitoring of local criminals is performed by local police. You could also say you’re monitoring the stock market, fashion world, or baseball standings if you follow any of them closely.