Digimon world 1 on the psx was the closest to the virtual pets, while injecting a rpg into it. You had to take care of your pet, and they had a short life span. In order to beat the game, you had to go through multiple generations in order to get the best digimon you could. You could screw yourself over pretty easily to and weaken your line. Overall though I felt it was the one closest to the shows given the story and focus on the adventure aspect of it.

Digimon world 2 is a dungeon crawler. You move a space, the enemy moves a space. It added a pokemon type mechanic in which you can capture digimon to add to your team. It also got rid of the raising a virtual pet aspect, and played like a standard RPG. I put something like 60 hours into this game without ever beating it. Loved it at the time, but would assume it has not aged well due to old mechanics and horrible graphics.

Digimon World 3, not my favorite. Rpg, but you fight one vs one instead of with your entire party. I really don't remember much else, it wasn't one that stuck out to me.

DS

Digimon dusk and dawn. Think of it as pokemon white and black, same game with slight differences. Plays similar to, but more....well annoying. The maps are hard to navigate, as the tiles don't actually line up correctly in some places. Combat is nice, but you only see the opponents and not your digimon. Think old dragon quest games. Still I really enjoyed the way they did this game. Evolving is up to you, and requires you to meet certain objectives for each form. Digimon have various evolution paths they can move down as well. It gave me smaller goals outside of the story to focus on, and I enjoyed that. There is also a virtual pet aspect to the game. You have various digivices you can set up with set land types and training devices to tailor it to certain types of digimon. I had a dark themed one, and a mechanical themed one for example. You then place digimon there to train while you are not using them. You have to make sure you have enough food production as well, so there is some enjoyment from messing around with the system.

There are a ton of other games, wonderswan had some fun ones. But those 4 are the main ones I would tell people about. Digimon World 1>Dusk/Dawn>DW2>DW3

They had 2 mmo's, they were fun to play around with. Not sure if they are still active. Total Cash Grabs though, you will not advance or get into the higher evolutions without paying.

They are also doing a remake of DW1 on the vita, sounds like it may get a western release. Sold my vita though, so I stopped following it.