Game genie -- change what the ROM looks like to the console. Game genies are the only things to do this.
Action replay -- change parts of the memory. Everything else (codebreaker, xploder, goldfinger*, game shark) is much the same.
*if you are speaking to a Chinese hacker this is what they will tend to translate it as, as most flash carts tend to come out of China it is worth knowing. Do also note it is what they also call the pins on a cartridge.
The only real differences are how they encrypt their codes if they are encrypted, how they encode their codes and what level of functionality is afforded by the device (there is much more to cheating than simple "hold this point in memory" type things after all).
Some consoles -- action replay/datel gear does nothing but allow you to inject hacked saves. The original xbox is probably the most notable example here though nowadays there are "trainers" which work much like action replay codes. Datel and co do also do a fine line in save dumping tools though and often on the same consoles that have more conventional cheating devices.
Bonus round -- the DS copies the entire binary (files are separate though the binary can include non binary code, I have fished every main type of data out of them over the years) to memory. This means you can change memory and change the binary or in effect make a kind of game genie code and definitely codes that could only otherwise be done by an ASM hack.
I have seen partway similar things for some Wii ROM hacks as well but that is a different discussion.http://doc.kodewerx.org/
is probably the most complete listing you will get for the various consoles. http://nocash.emubase.de/gbatek.htm#dscartcheatactionreplayds
are also worth a read though.
The PSP tends to be taken care of by CWCheat, most cheats are made using a kind of hardware debugging affair rather than emulators which are only just now getting to be useful.
"existing codes hacking a ROM"
Assuming you are not talking about the bonus round stuff this probably means master codes. They do various things but part of anti piracy protection and some instances of cheat protection is you check various parts of memory to see if things have been changed. Master codes will then tend to stop this protection from kicking in.
Otherwise they contain the memory location of a thing you are changing which can be very helpful. A while back I ported a cheat that forced the later DS Advance wars games to be in Japanese (it had Japanese in the ROM but would not load it from stock and was never released there). The cheat forced a copy of some of the firmware data to read 0 (signifying Japanese) but the game would not do it by default. As I was not about to make a mini cheat engine to force that area to be 0 I disassembled the ROM/ran it in a few emulators and I found the instruction that read that area of memory and forced the resulting routine to use 0 thus causing Japanese to run without ever touching that area of memory. That would be an example of how cheats could help, as I did not make the cheat I kind of suspect the cheat authors did a similar thing in reverse (my port was easy, that would have been a harder cheat to make).
Kind of related to that if you have an infinite items cheat you can use it as the starting point of a tracing session; you find something that reads the value you just made infinite, it might load the text description at the same time or as part of the related set of instructions, you now have a place to look for the text. It is a somewhat backwards method of going about it but not an invalid one or even one I could reasonably discourage.
That goes back to the quality of the cheat device thing more than complexity, if I am hardcoding button changes, hardcoding button activated cheats or some such then yeah I have to think about it but otherwise it is nothing major.
About the only thing I feel the need to add is the GBA, DS and to a less extent the megadrive/genesis were fairly simple in terms of the memory and how the carts were read (some aspects of the DS aside it was all visible at all points). The NES has mappers, the SNES has hirom and lorom as well as special chips, the GB/GBC has banks and it goes from there. This is not so bad once you understand what is going on though.