Terminate but Stay Resident was the original DOS function call to allow a utility program to exit but stay in memory waiting for interrupt signals (it will have hooked into those interrupts before calling TSR) to respond to. Utilities that operated in this manner (mouse drivers, and many other useful things) became collectively known as TSR programs and then just TSRs.

TSRs were a special kind of program, that instead of telling DOS "I'm done, here's my memory back", said "I'm done for now, but don't unload me", so their memory stayed allocated/used. And you got your command prompt back immediately to continue your work.

TSRs usually hooked into an interrupt like the keyboard interrupt, intercepting a hotkey to re-activate the program (so you could browse your address book while you were in your (text mode) word processor; "kind-of" multitasking), or they offered a service to other programs (like the mouse and CD-ROM drivers).

There were all kinds of things that ran as TSRs. Virus scanners (hooking into disk write/reads), modem tools (showing "RING" in the corner of the screen when the modem detected an incoming call), note-taking apps, address books, you name it.