What does "int 21h" mean in Assembly?

int 21h means, call the interrupt handler 0x21 which is the DOS Function dispatcher. the "mov ah,01h" is setting AH with 0x01, which is the Keyboard Input with Echo handler in the interrupt. See:

http://spike.scu.edu.au/~barry/interrupts.html

INT 21H will generate the software interrupt 0x21 (33 in decimal), causing the function pointed to by the 34th vector in the interrupt table to be executed, which is typically an MS-DOS API call.