First, you can try a restart. Hold down the Sleep/Wake button on top of the iPhone, wait for the Slide to power off message to appear - and then slide to turn the iPhone off. After the iPhone has switched off, wait a few seconds, then press and hold the Sleep/ Wake button again until the Apple logo appears.
If that doesn't work and your apps still crashes, try a Reset. You can manually reset your iPhone by pressing and holding both the Home and Sleep/Wake buttons for at least 10 seconds, or until the Apple logo appears.
If all else fails and you'll want to get your iPhone to its factory state. You'll need to do a Restore. You can only do this through iTunes. Connect your iPhone to your computer, and click on its name in the Devices list on the left-hand panel in iTunes. Click on Restore. All your precious information, such as contacts, bookmarks and so on, will be backed up, so you can reinstate them into your iPhone after the restore process has finished.
This might be because you do not have enough memory in your phone and you will have to delete some things that you do not really need. It might also be because your computer is not authorized and you will need to go to Store and authorize it in iTunes menu and then remove all apps and sync your phone again.
You have some great answers here, especially Jun's. There is one other possibility. If your iPhone is a 3G or older (not 3GS), the new update is causing your problem. You will want to take it to an Apple store for a deep restore, then only do the 4.1 or 4.2 update. When 4.0 came out last summer, it killed mine. Good luck.