The operating system in Android is Linux. To run an app with administrative privileges in Linux you have to run it under a program called su (or be logged in as the user named root - which is one of the top bad ideas of all time). Google left su out of Android (because it's powerful enough to allow you to delete Android itself).
In order to install su you have to have administrative privilege, which would seem to be a catch 22, since you don't have it until after you've installed su, which you can't, etc. Rooting is running an app or a program on a PC with the phone connected that uses some little bug or hole in the system to gain temporary administrative rights so it can install su.
Once you've done that (and added a couple of other programs, which a rooting program or app does), you can run apps that can only do what they need to do if they have administrative rights.
Programs that don't need administrative rights (and don't refuse to run on rooted devices - some do because the developer doesn't want his app running on a rooted device) won't see any difference. The phone won't be faster, it won't run with weaker signals, it won't give you more memory - it's the same phone unless you're doing something that needs administrative rights.