A Career is a long-term job that your Sim is able to advance in. If you Sim is good at their job, they can be promoted! Usually, the goal is to reach the highest level of the career your Sim is in.
The expansion Get To Work added Professions. Professions are careers that you, the player, can participate in. You can go to work with your Sim and do work related tasks, like helping patients in the hospital or finding clues as a detective. You can always send your Sim alone if you don't feel like going to work with them.
You can find the list of Careers available in The Sims 4 below.