Yakuza is a very well known series in Japan. From what I hear, the newest Yakuza game that came out in Japan is considered a PS3 seller. If you look towards the west, the Yakuza series is still relatively unknown. The latest Yakuza game to come out in the west is Yakuza 2.
Yakuza 2 is in essence a 3D brawler. The combat isn't deep at all at first and only adds depth later on with some new combos and techniques. Yakuza 2 is very combo-centric and sometimes Kazuma, the main character of the story and the dude you control, will keep punching air if you wiff a combo. You can also pick up weapons in this game to use against opponents. The weapons, while generally really effective against regular enemies, are useless against bosses and are cumbersome compared to regular hand to hand combat. The real highlight in the combat is what is called heat moves. Heat Moves are basically the equivalent of a super move in a fighting, once your heat gauge is full you can perform heat moves that do huge damage. Most heat moves look absolutely brutal when performed, something that never gets old.
Yakuza 2 also has some RPG elements to it. After every fight you gain experience points that you can spend on improving some aspect of Kazuma. These improvements range from increased health, learning a new technique, to doing more damage while your heat gauge is full. You can also equip Kazuma with weapons and armor before a match starts with some special effects. These are varied and are usually very helpful.
This should obviously go without saying, but don't go into Yakuza 2 expecting really great graphics. Even for a PS2 game the engine Yakuza 2 uses shows its age and even the PS3's upscaling doesn't help matters much. That being said, stylistic point of view Yakuza 2 is great. When you walk around the cityscape, it looks as though you are walking in a real city. Yakuza aims for a realistic yakuza feel and it does, all the people and locales fit within the feel. As far as presentation goes Yakuza 2 is almost at a MGS4 level as far as story-telling is concerned, it is just limited by it engine.
The story is really good and could easily be in a yakuza themed action-movie. If you missed the first Yakuza (like I did), Yakuza 2 puts you up to speed with the events that happened in the first game. The story is very much mired into general Japanese and yakuza culture, so all the Japanophiles out there will get an immediate kick out of this game. The only complaint I have with the story is that a couple of the final scenes have too many twists that were not really well executed. The story as whole more than makes up for this though. The reason I don't go too in depth of what the story is about is because it is that good.
If you just play through the story and pay little attention to anything else, the game is a tad on the shorter side of things. The story should last about 14-18 hours depending on how hellbent you are in finishing this game. To remedy this Yakuza 2 has many kinds of mini-games to pad out the time. The most infamous of these are the hostess bars where you can play a dating minigame. Most of these are hit or miss, and I didn't really bother with most of them since I wanted the story to keep on going.
Overall though if you want game with a great story that isn't the usual "Save The World from the Nazis/Demons/Aliens/Evil Twin" on the PS2, it is really hard to do better than Yakuza 2