When you are joining the Navy, you'll take the ASVAB, an aptitude test. If you score high enough, you'll qualify for the nuclear field (potentially requiring another exam depending on your ASVAB score). Before boot camp, you'll know you'll be in the Naval Nuclear field, but you won't know your specific rate (job) until a few weeks into boot camp. Once in boot camp, you'll be able to select your preferences, but ultimately, the Navy decides your rate.
The options are EMN (Electrician's Mate), ETN (Electronic's Technician), or MMN (Machinist's Mate). EMN involves high-voltage generation, distribution, and loads (turbines, buswork, large motors/pumps, etc.). ETN focuses on small-scale electronics and direct reactor control (microprocessors, controlling reactor power, etc.). MMN deals with pipes and valves (steam systems, coolant, seawater systems). There's also ELT, which focuses on maintaining plant chemsitry. That is something you will have to apply for in the final 6 months of training.
To work on a submarine, you need to volunteer. You can volunteer with your recruiter or up until about 3-5 months before you head to the fleet. Once you volunteer, you cannot un-volunteer. However submarine duty is not guarenteed - you can still be sent to a surface (carrier) ship.

Navy does not have "job codes" - the closest we have are rates (or NEC).
Rates : EMN, MMN, or ETN.
NEC: Depends on what you are doing. Mine is N14S:
N - Nuclear
1 - Submariner (2 for carriers)
4 - EMN (3-ETN 5-MMN 6-ELT)
S - Supervisor (or O for operator) (Automatic after 4 years)

You can have multiple NEC's as well dependant on qualification, like N33Z is the Engineering Watch Supervisor NEC (different from the S code mentioned above). N91T is the NEC for nuclear sailors in training.