DOS programs generally were like command line apps today (but simpler, obviously). They did a thing, then exited. Or they showed their "TUI" (as we'd call it today), and when you told it to exit, you got back to the DOS prompt to start the next program.

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.