IC/Microcontroller logic gates
There's no relationship, it's like asking about the relationship between cars and nuts and bolts - one contains the other, but there's more to it...

Have a look at www.nand2tetris.org which is a course starting at the basic building block (the NAND gate) and working up to a microcontroller playing tetris.

You can of course go further "back" than a NAND gate by breaking it down to its component transistors, etc., but that's probably not overly helpful to a software engineer unless the question is about the actual physics of system performance.