There is a Pokémon Center in nearly every town you pass through. The exceptions are your home town, where your house serves the purpose of a Pokémon Center (your Mom will heal your Pokémon, and you can store items in your PC in your room.)
Pokémon Centers have been seen in all the original Pokémon games. Starting with Pokémon Red/Blue/Yellow, Pokémon Centers served the purpose of healing your Pokemon, storing extra items in your PC, managing your Pokémon in your PC, rating your Pokédex with Professor Oak's PC and the Link Station (at the right-hand counter). When your Pokémon get healed, they get full HP and PP. In the games, Pokémon Gold/Silver/Crystal, the basic Pokémon Center got an update. While still having Nurse Joy with her Chansey behind the counter, the Link Station was moved to a brand new second floor. The PC still held Pokémon in Lists, but now, you have the option the re-arrange your Pokémon within each box.
The next update came with the games Pokemon Ruby/Sapphire/Emerald. Everything in the Pokemon Center received a makeover here. For starters, the Chansey that stood next to Nurse Joy is no longer present. In the PC, the first update you see are the Pokemon Boxes, which now look like boxes. Each box will hold five teams of 6, or a total of 30 Pokemon per. box. You no longer have to save every time you move a Pokemon, and there is a new feature, move items. The items part of the PC still remains a list of the items you have stored. You can also still get your Pokedex rated on the PC. You can also store any mail that your Pokemon is holding in the PC. The second floor is where the majority of the change occurred. The first station is an information lady, who tells you about the link station the first time you visit. The second station is the union room, which is a place where you can chat and trade Pokemon with your friends. You can also do a basic battle with Pokemon level 30 or less. The final station is the Colosseum (which is slightly misnamed). You can choose to battle with your friends, or you can mix records, allowing your friends secret bases to become accessible in your game. The next games with the Pokemon Center are Pokemon Fire-Red/Leaf-Green. Since the update to Ruby/Sapphire/Emerald, the Pokemon Centers stayed pretty much the same. The exception is the Pokemon Center on One Island. Once you get there, Bill sends you on a side-quest to obtain two gems to power a machine. Once your quest is complete, you will be able to trade between your Ruby/Sapphire/Emerald and Fire-Red/Leaf-Green.
The final games with a Pokemon Center are Pokemon Diamond/Pearl/Platinum. In the Pokemon Center here, you still have the PC, which now allows you to compare the stats of two different Pokemon, side-by-side. The largest change is the addition of an underground floor, used for Nintendo Wi-Fi connection. The station closest to the stairs is the information station. The next station is Wi-Fi communications, which allows you to communicate with any other trainer you have registered in your Pal Pad who is also online. The next station (only for Platinum) is the entrance to the Wi-Fi Plaza, where you can find three different mini-games to play, along with a few other activities. (There won't be a third station for the Wi-Fi Plaza in Diamond/Pearl.) The second floor also received a few updates. The window closest to the stairs no longer is only for information, you may now also sign the back of your trainer card. In the union room, you can now Draw, Chat, Trade, Mix Records, show your Trainer Case and do a basic battle. If you have Platinum, you can also do an egg swap. The final window is still a Colosseum, where most of the battle options remained the same.