monitoring is automated collection of a site or app's vital stats, e.g. is it up or down, how much traffic is it getting, what's its average latency, etc.
The appropriate amount and type of monitoring varies widely. Applications, for example, generally need more and deeper monitoring than web sites. For example, here's Bridgy's.
Monitoring is usually combined with alerting, which sends messages to the site owner or on call ops person when specific values change or exceed thresholds, e.g. three probe failures in a row. Often this is managed by an escalation system like PagerDuty.