How do I shine an LED when a PIR detects movement but keep it shining until a reset button is pressed?
The best answer is to use a small microcontroller, since there are inevitably more things for it to do.

However, the caveman method to do just this and nothing else is to use a set/reset flip-flop:



Follow the logic thru and you will see that this circuit has memory.  When IR goes low, OUT goes high.  With RESET high (not asserted), this makes the output of IC1B go low, which keeps OUT high.

RESET only needs to be blipped low for a short time to clear OUT back to low, assuming IR is not asserted at that time.

You can get such flip-flops already all made for you.  I show the inner workings to help you understand them.  Of course this can still be done with half a single quad 2-input NAND chip.