"Minecraft" is a game that encourages sharing your creations with other players. It can be annoying, then, to find yourself unable to connect to the server on which you have been building those creations. If you're dealing with an Outdated Client error, waiting for the problem to solve itself isn't likely to work.
Keeping up With Your Server
When you run into this error, the most likely scenario is one where your usual server has decided to upgrade. Many servers will go long stretches between major updates, while all of the bugs get ironed out of the latest release -- but you can't necessarily count on getting notice in advance. Try updating your client from the launcher before you connect again.
Dealing With General Outages
On rare occasions, the "Minecraft" authentication servers will go down. This can manifest is by returning "Outdated Client" errors. This is the rare situation where waiting out your problem will help. There is nothing you can do, besides wait for Mojang to get it together. This problem can be confirmed by checking the Mojang Support Center on their website (link in Resources).
Aside from simply updating to the most recent version of the game from time to time, some servers will choose to stay ahead by using the latest Beta release snapshot. That is going to make them much harder to keep up with, but it gets you early access to all of the newest features. If you want to try one of these servers, you can acquire and install snapshot releases from Mojang's website (link in Resources).
The Other Side of the Problem
Something to be aware of before you update your client in line with a server is the potential for mismatches elsewhere. Each release can only connect to servers that precisely match it. This means that if you upgrade to version 1.6.2 you will be unable to connect to a server running version 1.5. Keep that in mind before you jump at the chance to update. The error you will run into in this instance would be "Outdated Server."