Permanent root or temporary

Temporary root

These are usually one-step apps that you install on your phone, and they give you root access until the next time your phone is restarted.  They are a great way to get your feet wet,and you can do a lot with them -- both good and bad.  The ease of use makes temporary rooting pretty popular, and it's a fine choice if your reasons for rooting are to use root-enabled apps from the Market.

That being said, some apps just aren't going to work unless you go all out and permanently root your phone.  This depends a lot on which phone you're using, as manufacturers have an endless supply of dirty tricks to keep the hardware you paid for under lock and key.  You'll either have to ask users with the same model as you're using, or use trial and error.  The good news is that the popular root-enabled apps, and the ones you're most likely to want to use should work without a problem.  Titanium Backup,Wireless Tether, Root Explorer and the like should do just fine.

The last thing to keep in mind, is that not all changes will be permanent.  For example -- using a utility to "freeze" bloatware (applications from your carrier that you don't want or need) may not keep them frozen and hidden after a reboot.  Also some of the newer HTC phones have an evil little bug a feature that reverts any changes you have made to the system, bringing it back to the way it was before you started hacking away at it.  In those cases, the only fix is to perma-root your phone.

Permanent root

This is where things get a bit hairy.  Some phones, like the Nexus One, don't need to be rooted -- they can be unlocked via the Android SDK and modified at will.  Other phones, like the OG Droid, are really easy to root, and will only take a few minutes.  Finally, some phones, like the T-Mobile MyTouch 4G force you to jump through flaming hoops and follow often cryptic instructions, laden with warnings about bricking your phone.  That's a whole 'nother rant for a different venue, but it does need said so that you know what you may be getting yourself into.  You'll need to do your homework, ask any questions you feel are unanswered or are unsure of, and weigh all this into your decision.  Everyone who says "It's easy!" (including me) really means they found it easy -- that doesn't necessarily mean you will.

Now that I've sufficiently scared you, it's time to talk about why anyone in their right mind would go through this.  Besides the advantages of keeping changes persistent between reboots, and those few Market apps that won't work with a temp-rooted phone, you have what many consider the best part of owning an Android phone -- custom ROMs.

To flash any custom recovery, kernel, or ROM, you're going to need to have permanent root access to your phone.  Flashing new firmware is very low level stuff, and you need read and write access to everything.  The good news is that usually the rooting is the most difficult part, and a custom recovery gives you access to a tool that can take a snapshot of your system, and save it as a restore point.  That's a good thing, because once you start flashing, you're hooked -- and a single command restore is priceless.