I will try to answer this one, but I am not an accountant, and it is a little complicated.
Suppose that your job entitles you to a pension.
However, you only get the pension if you work five years. If you don't work at least five years for the company, then you get nothing.
After five years, your pension "vests," in other words, you have possession of it and will get it when you retire.
The word is also used in the context of stock options. If you get a stock option, you have a chance to buy stock at some point in the future. You might be promised stock options at the end of every year, for example. However, you can't buy the stock until the option "vests" at the end of the year, and if you leave the company before the option is "vested," then you don't get it.
Basically, it can apply to the moment you take ownership of almost any kind of property.
That's my best understanding of the word. I hope that helps.