Before I answer the question you asked, let me answer one you didn't: Should I install a task killer?
A: What is your goal in doing this? What do you think it will get you. In general, task killers aren't needed on any version of Android newer than 2.0. In fact, Task Killers often make things worse, and if not configured properly can cause instability, force closes, reboots, and increased (rather than improved) battery use. Many people seem to think that having unused apps taking up memory somehow impacts performance or increases battery usage, but that's really not true. Android keeps recently used apps in memory, in an inactive state, on the assumption that if you've used it recently, you'll possibly use it again soon. It's much faster, and much more efficient, to simply access in RAM than to have to copy it from storage (ROM) back into RAM, reinitialize the app, and then redraw the screen. RAM that's "filled" with an inactive app uses no more power than "empty" RAM. Every bit of RAM is constantly refreshed, whether it's "used" or not. And if a new app needs that RAM, it's only marginally less efficient to load it into "used" memory than "empty" memory. Essentially, Android just has to report to itself that the app that used to be there isn't there any more. The write operation is the same either way.
Some people think having all these apps "running" in memory slows the system down. That's true if the apps are really active, but for the most part Android apps in the background are just waiting, not actively using CPU. And those apps and services that ARE using CPU in the background are probably important system processes. Killing them will cause the instability I referred to previously. If you have a specific app that keeps running in the background when it doesn't need to, chewing up CPU and battery, you should find a replacement for that app that is properly coded.
In short, don't use task killers. Let Android do what it was designed to do. It really does it very well. If you insist on using a task killer, you don't need to be rooted to do so. There are apps that will do this without requiring root.
Now then, what is rooting:
Get 'Root' on your Android device | Android Central
Rooting - is it for me? Some Q&A | Android Central
http://forums.androidcentral.com/ve...ing-roms-hacks/205601-what-do-after-root.html
http://forums.androidcentral.com/t-mobile-galaxy-s-iii/217043-what-can-i-do-rooted-phone.html
In most of the carrier forums under this S3 forum, you'll see a rooting sub-forum. Read through those forums, paying special attention to any post that includes the word "brick." My phone is rooted, but I've been at this for quite a while. The S3 is an easy phone to root, IF you use a method that is specific to your version of the S3 (A t-mobile S3 <> a Verizon S3 <> a Sprint S3, etc.) and IF you follow those instructions very carefully. It's also possible to screw things up if you do it wrong. Do note that "rooting" and "flashing ROMs" are two different things. You need to root in order to flash ROMs, but you can root without flashing a ROM. Flashing ROMs are where things get really dangerous, if you flash the wrong one.
My suggestion: if you don't know why you need to root, or none of the reasons people give for rooting are compelling to you, don't. The S3 works great as it is. Rooting voids your warranty. Rooting prevents you from getting updates directly from your carrier (although those updates are generally rooted and made available here pretty quickly). And rooting makes it easier to screw up your phone, get hacked, or otherwise have issues.
I don't mean to come across as too negative. If you have a good reason to root, we'll help you do it safely. Just don't do it for the sake of doing it, or because "everyone else" is doing it.
HTH. Feel free to ask any other questions you might have.
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