What are they and how do you get them?

At your pet profile, underneath level up points, there's a "proficiencies" button, that will take you to the pet's proficiency tree page. Proficiency points are basically extra points you can use to boost your pet's stats/fighting ability. The different trees matches different stat profiles (strength vs intelligence, dexterity vs agility), and also gives extra abilities. Some abilities can heal the pet, others can cause the enemy more damage.

I'll recommend putting your proficiency points in one of the stats, instead of an ability. Proficiency points stack up, so when you put a lot in the same stat, you actually increase the pet's stat way more than if you put the same amount of level up points in that stat. But that's a personal preference.

You get 1 proficiency point for every 5 levels your pet gain.
(So you can only get a max of 12 points now, and 15 points when the megazone is released and the level cap is raised to 75)

Yeah, aren't you supposed to be able to end up getting 50 points?

The 50 at the bottom of the proficiency tree window is the level is the projected max, but the ability and content to get there hasn't been programmed yet. ;u;

We might not be able to get all 50 points until a few years from now, when the rest of the game's content is programmed and unlocked.