Pretty open-ended question, but here are a few things I have picked up.
For super simple processors, there is no true startup code. The cpu gets power and then starts running the first instruction in its memory: no muss no fuss.
A little further up we have mcu's like avr's and pic's. These have very little start up code. The only thing that really needs to be done is to set up the interrupt jump table with appropriate addresses. After that it is up to the application code (the only program) to do its thing. The good news is that you as the developer doesn't generally have to worry about these things: that's what libc is for.
After that we have things like simple arm based chips; more complicated than the avr's and pic's, but still pretty simple. These also have to setup the interrupt table, as well as make sure the clock is set correctly, and start any needed on chip components (basic interrupts etc.). Have a look at this pdf from Atmel, it details the start up procedure for an ARM 7 chip.
Farther up the food chain we have full-on PCs (x86, amd64, etc.). The startup code for these is really the BIOS, which are horrendously complicated.