I'm going to use a shift register to turn off and on some LEDs. However, I just read on the M74HC595 datasheet that it can only output 6mA on each pin. The LEDs I got are rated at 20 mA. Do I need the 20mA to power the LED or is 20mA the limit?
Leds are rated for a given Current for ""full"" brightness for a specified life time. Most "regular" leds are 20mA typical, for x number of thousand hours. They can be driven at a lower current for extended life, or at a higher current at the cost of life (There are maximums to how much current can be used, as the current causes heat which then kills the led).
You can, and often should drive them with less current, unless you need full brightness. Just use Ohm's law to get the correct resistor for the 6mA current.
(Voltage Source - LED Forward Voltage) / Current (In Amps, so 0.006).
For the most part, leds at full brightness can be obnoxiously bright at straight angles. I've gone in to change the resistor for some gameboys because those things will blind you. 6mA is plainly visible in room lighting.
It is hard to say. What you should do is power an LED with 6 mA and see how bright it is. Most LEDs are designed for 20 mA but will light up dimmer with less current. How dim depends on the LED.
If you need more than 6 mA then you either need a different shift register, or you need something else to drive the LEDs (like an LED driver or some transistors).
There are many shift registers that are designed to drive LEDs with a constant current. Here is one that I randomly picked from T.I.. There are dozens available from TI and other companies.