Persistent browser-based games are actually simple text-based browser role playing games. The short term for persistent browser-based games is PBBG. People usually prefer text-based games term over PBBG, but the correct term is still persistent browser-based games, so keep that in mind.
First of all, grinding is the phase of the game where you just have to do the same repetitive moves over and over again. And yes, every time you log in you do the same things over and over again. But what makes it more interesting is that you can compete against others. You want to be number one in the ranking, don’t you? Don’t lie, yes you do. So you don’t really care if the game itself is repetitive as long as you are one of the best players in the game. And the only way you can beat others is just grinding and investing a lot of time into your character’s progress.
Some call their text-based browser games something like MMOG, which stands for Multi Player Online Game. Some even say browser-based RPG. Others just say text-based browser games. It really is up to the game developers how they advertise their game. Current leader is Torn City.