I'm beginning work at a firm with an incrementally vested retirement pension (not 401K) plan that is incrementally vested for 20% for each of the first five years. After five years, I'm 100% vested. What does this mean for my pension if I leave the firm after 3 years?
Vesting generally means that you are entitled to a pension as per the plan rules. If you are not vested and your employment is terminated (e.g. via resignation or firing), the pension plan will just return all the money accumulated thus far to you (possibly roll it over into your IRA or your new employer's pension plan or 401k plan if that is possible). When you are fully vested, you have the choice of whether to take your money with you upon termination of employment or leave it in the pension plan, hopefully to grow, and take a pension when you ultimately retire. So, if the pension plan rules say that you will get x% per year of your final salary (usually x% is between 1% and 2%) then after five years when you are fully vested, you have the right to get 5x% of your salary starting from the full retirement date (as stated in the pension plan rules).
Another meaning of vesting is about 401k plans where you cannot take the employer match with you if you quit or are fired before full vesting occurs. Your own contributions are yours to take at any time, but the employer's contributions can be taken only after vesting.
Partial vesting means you are entitled to the fraction vested of the full benefit. For example, for a 401k plan, after one year you are entitled to take 20% of the matching funds contributed by your employer, after two years, 40% etc. Remember that over 5 years your employer will likely contribute increasing amount, but even if not, you will be entitled to 20% of first year match, 40% of two years' match (including earnings), 60% of three years.... etc.