A monitor is a device or a person who watches or checks to see that all is going well. If you're babysitting, you might use a baby monitor so you can hear if the baby cries in its sleep.
Monitor can be used as a noun or a verb, both of which are concerned with watching over someone or something. A heart monitor displays the rhythm of your heart. A hall monitor is a person who checks to see that students aren't wandering the halls. If you are monitoring your grades, you are keeping track of them to see if they go up or down. And some parents buy a program that will monitor your Internet use, though you might think that this is an invasion of privacy.