Ok, I went from an average of 25 FPS all the way to 350 with a few simple steps. Follow these and watch your FPS raise at least a little bit.

1. Hit Windows+R or search run in the start menu. It will pull up a small box, like the one below. In it, type "msconfig" without quotes. Hit enter.

2. Go to the Startup tab and hit the disable all button. This disables all processes on startup, but Windows will know what it NEEDS to boot up and it starts processes as they are needed, rather than starting everything, some of which you most likely never use.

3 (optional). IF you do not use windows media player network sharing, definitely do not skip this step. This process used up 25% of my CPU on average, with spiked hitting up to 30%. That's just not necessary.
3a. Go to start and type "services.msc" without quotes and press enter.
3b. In the Services tab, look for the entry called "Windows Media Player Network Sharing Services."
3c. Double click that service and set the Startup type to Disabled.
4. Know how much RAM your PC has. To find this, go to Control Panel > System > Installed Memory (RAM):
There it will tell you how much RAM you've got installed.
4a. Open the Minecraft Launcher and click Edit Profile.
4b. At the bottom of the box that popped up, enable JVM Arguments. In there, you will see something with -Xmx in the beginning. I've got 16 gigabytes, so I give Minecraft 1/4th of my installed RAM (4gb). For example, to assign 4gb of RAM, the Argument in the box will be "-Xmx4G" without quotes. Save your profile.
5. Restart your computer. Notice your startup speed will be faster.
6. Launch Minecraft and look at your FPS.
7. Restart your computer at least weekly.
8. Post your results for others to see.

Note: I have 2 mods installed: Forge ModLoader, and Zan's MiniMap. I'm not sure whether they affect my performance or not. I am not running OptiFine.

Also Note: This boosts your FPS in more than just Minecraft. I was unable to play Battlefield 3 because I was getting such a low FPS in it, now I get roughly 120 on medium graphics, and I can play again!

Enjoy!

(Here is my original post about this.)

UPDATE:
A note for people running windows 8:
Windows Media Player Network Sharing Services will be marked as WNPNetSvc in Task Manager service tab. So disable that. (I have not confirmed this, only been told my another user of Windows 8.)