[Galaxy S III General (US Carriers)] Raegx:
This is a hard question to answer without just listing of the capabilities of applications that are available on Play. I will just mention what I know in a very basic way. It isn't all 100% technically correct, but if you know enough to say it isn't 100% then my explanation is probably not for you. =D

There are system level changes that can be made to your phone that are normally protected. Android, under the hood, has a file system and for all intents and purposes is very similar to what you see in Windows/OSX. It however, does not have a pretty graphical UI over it that is exposed to you by default.

Most of the system's settings and applications are stored very specific directories on the file system and by default you do not have the ability to directly change anything in those directories; this is done via permissions if you care to read more about it. When you are running your phone you are essentially running as a restricted user. When you root your phone you are giving yourself access to the 'root' user. On a Linux system that is essentially the account that is allowed to change and access anything on the file system; thus all the previous restrictions I mentioned are gone.

This gives you many abilities, some of which are trivial and some of which are not. Consider that as your non-rooted aka restricted user, you can not remove 'bloat ware' from your phone. Verizon is very notable for doing this. You essentially are not allowed by permissions to do so. However as root, you can delete whatever you want. Applications in Android are binary files with the *.apk extension and can be roughly (not always correctly) deleted by removing the *.apk file. You can also do things like directly read the *.apk files and upload them to another site as away to back up your applications (Titanium Backup). There are also more esoteric things like changing some of the settings file where where buffer sizes are kept for reads/writes, etc (SetCPU and flash 'overclocking' apps).

There are more such as wireless tethering (allowing your phone to act as an Wireless Access Point so your wireless desktop/laptop can use your 3g/2g/4g/etc data connection on the road), customizing the look and feel of the ui, etc.

Yes and no, but mostly yes. I don't want to scare you, but rooting your phone opens your phone up to higher access by you and by the applications you install. You do have the ability to deny applications super user access, but if you do you do need to realize that super user granted application will have unrestricted access to your phone and data. So only do it for applications where you can verify the application your self or trust the publisher.

The actual root process is generally 'revertible'. By revertible I mean that you will be able unroot/get your phone into a working status, but you will most likely lose any data that is stored only on your phone. You can however back data up to other sources prior to doing anything potentially risky. By the time a guide is posted for a newer user to follow, the process has been tested to a level where it won't outright damage your phone on physical level. Usually anything wonky can be fixed by flashing a stock ROM (which yes does include losing all of your data unless you back it up somewhere). Be aware that the first time you do this you will probably be looking at investing time into learning jargon and programs you've never heard of before. It can be daunting, but most of the guides are posted in step-by-step format.

They will have the same abilities. Some stock ROMs come with carrier (e.g. Verizon, Sprint, etc) applications that will most likely not be present on the custom ROM. However most of the time if you really want them you can still in stall them and they will work. I am not currently aware any that don't.