It stands for "Terminate and Stay Resident". In DOS, you can only have one program running at the same time. TSR programs are a workaround. If you want more than one program to work together, you have to have the first program run until it's done everything you need, leave its variables/memory somewhere accessible for the second program, and then terminate to let the second program run.
It also stands for "The Screening Room", a weekly movie reviews show that I work on here at Ithaca College. It's live on ICTV Wednesday nights from 8:30 to 9:00, and reaired twice during the week.