Yes, crystals are often used to generate clock pulses, especially when accuracy is important. If accuracy is not critical, lots of other ways to generate clocks: 555 timers, multivibrators, phase-shift oscillators, etc.
All chips like microprocessors require a clock to operate: otherwise, they'd just be sitting there in a static state doing nothing. The clock drives the operation of the chip (or circuit), advancing it from one state to the next. The state may or may not output anything. In a chip like your computer's CPU, the clock steps the processor through its "microprogram", where each machine instruction (like, say, ADD A B, which adds the contents of two registers), may take several clock cycles to complete.