People use the term "vesting" to refer to a clause typically embodied in a founders' agreement in which the shares "vest," or become shares that you actually have a right of ownership over, if you continue to perform your duties.
Vesting schedules typically last 4 years with shares vesting quarterly or monthly.
So, you can think of it like this: After year one, you'll own a 1/4 of your 50%, or 12.5%. Year two, 1/2, or 25% ...and so on. You don't actually own the full 50% (i.e. doesn't "fully vest") until year 4.
Many vesting agreements carry a one year cliff, which means that none of your shares vest if you leave the company before performing for one full year. This is to prevent the "lazy founder" problem, where one founder is doing most of the work and the other one is goofing off, or decides to leave and take another job.
This is important because the first year is the most risky for a startup and requires all hands on deck. If someone flakes out, you can't have a big chunk of shares tied up in them. One, because it isn't fair. Two, because you likely need to go find yourself another founder or will need that equity for first employees. It scares off experienced founders and investors to see a chunk of your cap table trapped in a "bad decision" founder that flaked out. This same logic carries over to why there are vesting arrangements... you shouldn't be compensated if you abandon the company and don't put in the same work as the other founder.
Hope that helped explain what it is and the reasoning for it!