I see this question over and over and I've responded to it many times on places like Yahoo Answers so I thought it was time to write a quick guide.
Step 0: (since I forgot to mention this on the first edit) Backup any important documents or information. Updating your device will erase all saved data and applications you've installed. It basically restores it to an "out of the box" state.
Step 1: Determine the make (HP, HTC, Toshiba, etc) and model (111, G900, Touch) of your phone. You need to know exactly what phone you're using before you update.
Step 2: Determine which version of Windows Mobile you currently have. Open "About" under settings. On most devices its written out in plain english - ie "Windows Mobile 2003 Second Edition".
Step 3: Go to the manufactures website to find the latest update. Sometimes companies will give you free updates, sometimes they're paid upgrades .. and sometimes they just don't exist.
Step 4: Read the instructions carefully!!
Step 5: Follow the instructions and update your device.
Now, if you can't find an official update - there still may be hope. For many devices, you can get what is known as a "cooked ROM". Essentially this is an update created by hacker enthusiasts and is NOT supported by the manufacture. If you have a warranty installing a cooked ROM will almost certainly void it - and you run the risk of damaging your device in the process. I've been experimenting with cooked ROMs for years and have never had a problem - but you're still installing at your own risk. If the installation fails part way through it could permanently ruin your phone.
If you want to try a cooked ROM start researching on google. Another excellent place to look is on XDA developers - there are cooked ROMs available for many popular Windows Mobile devices.
One last thing to note - when installing an unofficial or cooked ROM, you'll almost certainly have to install it from a memory card. So make sure you have one handy - and its very recommended to not put any files on the memory card that aren't essential to the update/upgrade.
Good luck and read the instructions!