An "alert message" is a warning on a computer or other machine. It tells you about something that you should pay attention to.
One place that you might find alert messages is on a bank website. If you're logged in for a certain amount of time without any activity, an alert message will pop up and ask you if you want to stay logged in.
You may also find alert messages on your phone when the phone's battery starts to run out.
Similar to "alert messages" are "warnings", which are a little more serious, and "notifications", which are less serious.