There are some apps that will only work on a rooted phone. Titanium Backup is one. Others include SetCPU, which can help with battery life and overclock some phones; Roam Control, to let you force roaming mode when you have a weak signal on your carrier; AdFree, and similar programs, to block ads on web sites and in some programs. There are many others. Rooting lets you tether another device to your phone to let it access the internet using your 3G or 4G connection (without having to pay the carrier's surcharges for tethering). Some apps provide additional capabilities on a rooted device: ES File Explorer, for example, will let you access system files, which can be very dangerous, but also a very powerful way to make changes to the way your phone works. (On Sprint GS3's with Jelly Bean, for example, this is the only way to eliminate the shutter click sound when taking pictures).
3rd party ROMs vary from completely stock Android, without various vendor add-ons (such as TouchWiz or Sense), to heavily customized versions with a dramatically different theme (look and feel), new programs built in, other programs removed, and various other tweaks. Note that many of these 3rd party ROMs are buggy; some tremendously so. Flashing new ROMs to your phone tends to be a "hobby" more than a real need.
When Joe asked where you get "the things", I think he meant ROMs, not applications.
WARNING: Never, ever, even think about flashing a ROM that isn't specifically tailored to your exact phone model and carrier. Flashing an AT&T ROM (as just one example) on a Sprint phone is almost guaranteed to create a very expensive door stop.
If you look at the carrier specific forums here (within the SG3 forum), you'll see a rooting subsection in each. There are discussion threads about various ROMs, and you'll find links to the ROMs in the threads. If you ask in the rooting forum for your device / carrier, you'll get some more specific answers. Probably the ultimate source for information about different ROMs is XDA-Developers:
Android, Windows Phone, and Windows Mobile Development News, Information, and Howtos - XDA Developers Take a gander through the forums dedicated to your device. Be prepared to have your eyes glaze over, though.
Finally, and this is just one man's opinion: If you don't know why you want to flash a different ROM, don't. If your phone is a tool, rather than a hobby, just say "no." Flashing a ROM isn't rocket science, but neither is it like installing just another program. It is possible to mess up your phone pretty badly if you don't do it right, and while it's usually possible to recover, there's always a risk of bricking your phone. I've flashed ROMs on other devices to add capabilities that the standard ROM didn't offer. But Jelly Bean is a pretty strong from a features and useability standpoint, and already has what those earlier custom ROMs offered, and more. I plan to keep the stock ROM on my GS3. Maybe something will come along later to change my mind, but right now I don't see it.
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