Email
Alerts go out the moment a check fails, and again when it recovers. By default they go to your account address; each monitor can have its own list instead.
Set a monitor's recipients deliberately if it is a test monitor. A monitor with no list alerts the account owner, which is rarely what you want from something you are deliberately breaking.
Webhooks
Each monitor can fire a webhook on failure and recovery. That is how Slack, Discord, n8n or anything else that accepts a POST gets told.
The webhook carries which monitor, what happened and when, so a receiving system can decide what to do without calling back.
What is worth alerting on
Uptime is the obvious one and the least interesting. The ones that catch real problems are certificate expiry, domain expiry, and heartbeats that stop arriving — a site that is up but not sending mail looks fine to a homepage check and is completely broken to your members.