Imagine you're working as an administrator of a large IT infrastructure. You just started receiving e-mails that a web application just stopped working. When you try to access the same page, it just doesn't load. What are the possibilities? Is it the router? Is it the firewall? Perhaps the machine hosting the page is down? Before you even start thinking rationally on what to do, your boss calls about the critical situation and demands explanations. In all this panic, you'll probably start plugging everything in and out of the network, rebooting the machine…and that doesn't help.
After hours of nervous digging into the issue, you've finally found the solution: the web server was working properly, but it would time out communication with the database server. This was because the machine with the DB did not receive the correct IP as yet another box ran out of memory and killed the DHCP server on it. Imagine how much time it would take to find all that manually? It would be a nightmare if the database server was in another branch of the company or in a different time zone and perhaps guys over there were still sleeping.
But what if you had Nagios up and running across your entire company? You would just go to the web interface and see that there are no problems with the web server and the machine on which it is running. There would also be a list of issues—the machine serving IP addresses to the entire company does not do its job and the database is down. If the setup also monitored the DHCP server itself, you'd get a warning e-mail that little swap memory is available on it or too many processes are running. Maybe it would even have an event handler for such cases to just kill or restart noncritical processes. Also, Nagios will try to restart the dhcpd process over the network in case it is down.
In the worst case, Nagios would speed up hours of investigation to 10 minutes. In the best case, you would just get an e-mail that there was such a problem and another e-mail that it's already fixed. You would just disable a few services and increase the swap size for the DHCP machine and solve the problem once and for all. Nobody would even notice that there was such a problem.