In the Mario Brothers universe, We always hear about the princess Peach and the Mushroom Kingdom, but are there any other kingdoms in the Mario Brothers universe?
According to the Super Mario Wiki there are quite a few!
If we stick to just main series Mario games, you have:
the Beanbean Kingdom from Mario and Luigi SuperStar Saga.
Sammer's Kingdom from Super Paper Mario.
Flora Kingdom mentioned but not actually see in Super Paper Mario.
The Dry Dry Kingdom mentioned in Paper Mario, but not actually seen.
Sprixie Kingdom from Super Mario 3D World and Toad's Treasure Tracker.
Nimbus Land from Super Mario RPG, which although it is called a Land and not a Kingdom, does have a King and Queen.
There are also various Lands, Towns, Towers, Forests and other areas not listed here, as you were only asking about Kingdoms specifically. Donkey Kong Jungle Beat has various fruit themed Kingdoms as well. There is also the Baseball Kingdom mentioned in Mario Super Sluggers.
Also as pointed out in comments, if we believe Luigi when he mentions it in Thousand Year Door, there is also a Waffle Kingdom.
Yup, there are.
Super Mario Galaxy 1 and 2 introduces many new planets, and by extension many new kingdoms/land/countries/bakeries/et cetera as well. Unless the Mushroom Kingdom recently became a galactic empire, I think it's safe to say all of these planets are located outside the Mushroom Kingdom.
Mario is Missing takes place..well, all over the planet earth. The first Mario Golf game takes place on the planet Earth as well. There's also the Donkey Kong games, which was likely intended to take place in Earth as well (I don't think the Mushroom Kingdom was even conceived until Super Mario Brothers).
I'm sure there are more I've missed, but as you can see, there's a lot more to the Mario Universe than just the Mushroom Kingdom.
In Mario and Luigi Dream Team, we go to Pi'llo Kingdom.
Bowser is the Koopa King (or King of Koopas, if you prefer) and has his own castle with subjects that he rules over. Though Super Mario 3 shows us a large, spooky, highly-militarized "world" that Bowser rules over, it is referred to as "Dark Land" and not as a Kingdom of its own.
However, Bowser is the only known actual king* in the main Mario series, outside of RPGs and sports games, so if anyone qualifies for ruling over a Kingdom, it'd be him.
*King Bob-Omb and King Goomba are both trapped in paintings, so whether or not they actually exist is...questionable.
In Super Mario Odyssey we’re introduced to the Cap, Cascade, Sand, Lake, Wooded, Cloud, Lost, Metro, Snow, Seaside, Luncheon, Bowser’s, and Moon Kingdoms.
Super Mario World for the SNES takes place in Dinosaur Land. Mario, Luigi, and Princess Peach go there on vacation and King Koopa kidnaps the Princess again.