Logic modules compare two inputs, then make an output according to the two inputs. Logic modules (usually) only react to high or low ouputs, often 0 or 5V - with a tipping point somewhere in the middle - and also only output 0 or 5V. We usually call these 0 and 1, even though it's not 1V it is called one because logic gates are basically really simple 1 bit calculators. They are binary
There are three basic logic gates, each of which also have an inverted twin brother:
AND - If A and B are 1, then output is 1. If only A or B or none are 1, then output is 0
OR - If A or B are 1, then output is 1. Also 1 when A and B are hight. If neither are 1, then output is 0
XOR - If A or B are 1, then output is one but not if both are high -- the X is for eXclusive, it has to be exclusively one or the other but not both.
The three twin brother gates are NAND, NOR and NXOR which are exactly the same, but the output is opposite. So NAND gates output 1 until both inputs go high, then it goes low -- opposite to AND.
So you are maybe wondering how this is useful and there are actually a ton of ways. Gates and clocks are binary signals and are the perfect inputs for logic gates. Here are some ideas:
You can use an AND gate to switch a clock signal on and off -- run the clock into A, use a gate signal at B to gate(!) the clock signal. The output will be the same as the clock, but only when B is high!
XORs can be used to "square" up audio, and obliterate it into a nasty square wave. Just use one input. Very aggressive. Even better, feed two audio signals into the same gate and they ring modulate one another! Ugly, lo-res, retro arcade machine aliasing sounds.
Create a third baby rhythm from a mummy and daddy rythym. This works with any of the gates. Feed two gates or clock signals into a logic gate and the output will be something related to both of them. I love XORs for this, you can create some really interesting patterns.
Comparing things (you might need comparators for this). How about having a bassline that only comes in when two LFOs are high enough? Simple - run the LFOs into comparators, then run that output into and AND gate, then send the AND output the CV input of the VCA the bassline is going too.
I hope this gives you a decent idea of what they can be used for. Incredibly powerful tools for sequencing and controlling things, maybe not quite as useful for keyboard jamming.