Hit a webserver with many thousands or tens of thousands of requests per second, and whatever the weakest link is will break: memory usage, CPU usage, database, bandwidth, billing, third-party service quotas...
At the time when the Slashdot effect was at its peak, the limiting factors were probably RAM or bandwidth (third-party services was not as much a thing, and CPUs were generally able to push 10 Mbit/sec without much of a problem -- but 10 Mbit/sec Internet lines would be a luxury). Like, you could reasonably host Apache on 16 MB of RAM on a 2 Mbit/sec line for your hobby site with a couple of photos of your car or whatever, and then ten thousand people would come along at once and find it interesting, and your site would be dead for a while.
