The reason I ask is that Stack Overflow has been Slashdotted, and Redditted.
First, what kinds of effect does this have on the servers that power a website? Second, what can be done by system administrators to ensure that their sites remain up and running as best as possible?
Unfortunately, if you haven't planned for this before it happens, it's probably too late and your users will have a poor experience.
Scalability is your first immediate concern. You may start getting more hits per second than you were getting per month. Your first line of defense is good programming and design. Make sure you're not doing anything stupid like reloading data from a database multiple times per request instead of caching it. Before the spike happens, you need to do some fairly realistic load tests to see where the bottlenecks are.
For absurdly high traffic, consider the ability to switch some dynamic pages over to static pages.
Having a server architecture that can scale also helps. Shared hosts generally don't scale. A single dedicated machine generally doesn't scale. Using something like Amazon's EC2 to host can help, especially if you plan for a cluster of servers from the beginning (even if your cluster is a single computer).
You're next major concern is security. You're suddenly a much bigger target for the bad guys. Make sure you have a good security plan in place. This is something you should always have, but it become more important with high usage.