Answer:
Unlike Enlisted ranks, Commissioned Naval Officers advance by a different set of standards, though time at current rank is one thing that both have in common.
Officers, depending upon their career path (Line Officer, Engineer Officer, JAG, Medical, etc.) must fulfill certain responsibilities, as well as have the support and recommendation of their Commanding Officer. Evaluations are much more important for rank progression for Commissioned Officers than they are for Non-Coms, in which Evals are only part of the overall advancement percentage. Officer Evals are essentially what makes or breaks whether or not they progress or not. A single mistake can literally ruin an Officer's career - I've seen it more than once. For example, on a Nuclear Powered vessel, mistakes of any kind are not tolerated - period, and on the rare occasion they happen, they are dealt with severely. I remember a Lt. who missed a clerical mistake on a tag for an engine room component. The resulting letter in is record ruined any hope he had for advancement.
Of course the fact that the error was caught be a Naval Reactors inspector on a surprise audit didn't help his situation much either....
Fitness evaluations (FitReps, or Fitness Reports) are also a large part of an Officer's advancement. Each service record includes a full top-bottom picture of the Officer in question, in full uniform, standing at attention. The Navy does not want slackers in command, and Commanding Officers are who the public will see if that ship is someday in the news for whatever reason. Unfair? Maybe, but there's a lot more to it than you might think. While it's technically supposed to be as important as the overall evaluation, the FitRep is in fact one of the biggest factors for advancement. Why? If you're not able to maintain yourself and your own fitness, how are you going to be able to command a vessel or shore installation? Would you want someone who can't control his own habits in charge of a nuclear powered warship with nuclear weapons? Standards are very stringent for certain career paths, and they need to be for a very good reason.
Being in command means being perfect, or at least appearing to be. A Captain of ship at sea is God, literally, to those on board, and the Navy cannot afford to promote those who aren't capable. While the evaluations and selections might at times seem biased, (it's a fact that Academy graduates get better positions than ROTC grads do) the system has proven that it works.
It's a bit different for non Line Officers - JAG's, MO's, Supply Officers, etc., those that aren't command responsible for the most part. While there of course are those in those fields that command, there aren't as many as there are Line Officers, and the Navy isn't as concerned, at least in most areas. In those areas where the command might be of an extremely critical nature, you can bet the requirements are every bit as stringent as they are for Line Officers.
So -
1. Time in current Rank
2. Responsibilities/Command/Supervision at that Rank
3. CO / XO Recommendations & Evaluations of the Officer at that Rank
4. The Officer's FitRep.
5. No negatives in the Service Record. It only takes one to derail a career.