I am very attached to Team Rocket and Team Galactic.
I see a lot of similarities between the two. I enjoyed both bosses, Giovanni and Cyrus, they were both very active and working hard for their organizations. Their personalities are very well developed and they pull you into their stories. They make me very curious about their background and thought process. They also have some very lively underlings that work for them. When I was a child, I always loved Jessie and James. When Diamond and Pear came out, my best friend and I absolutely adored Mars and Jupiter.
I also like how individualized they are. Team Rocket are just out for money. They don't care who gets hurt or why people are upset. If they take a Pokemon from someone, much like Galactic, they don't see the point in a trainer fussing over it, because they can always catch new ones. They were so into their duties that even killing a Marowak and orphaning it's child didn't stop them. They were always so epic to me.
When it comes to Team Galactic, they were out to create a new world. I love that they didn't like how things were run in this world, and they were out to make something better. A perfect world of no strife. And they were so obsessed with space. Cyrus bolted into his new world on top of Spear Piller with not even a second thought. It was gold. Mars, Saturn, and Jupiter were pretty epic too, I found their personalities to be really developed.
The other teams, especially Team Plasma, just didn't catch my interest. I thought Aqua and Magma were so-so. I loved the Magma outfits. But that's it.