The Game Corner has been a staple of every Pokemon game up to Generation IV. Here you can buy coins, test your luck, and win fabulous prizes, including new Pokemon and TMs you won't get anywhere else.

The Basics

You'll need a Coin Case, to be found in the restauraunt from the guy with the glasses along the main strip of buildings below where the Rocket Game Corner is. At the main desk in Game Corner (in the top left corner) you can buy 50 coins for 1000P. Unlike more recent titles, there is no 500 coins for 10000P option: you're stuck mashing the A-button unless you decide to play a few games. Make sure you look around everywhere on the ground in the Game Corner: there are plenty of free coins other people have dropped, to be found as you would with any other hidden item. Your Itemfinder won't pick them up, though.
Slots and Games

The only game in the casino is a slot machine-style game. Insert however many coins you want (1, 2 or 3, which add more rows on the slots where you can win prizes) and try and line up the same picture. A triple 7 yields 300 coins, three BAR yield 100 coins, three Poliwag, Diglett, or Jigglypuff win 15 coins, and three cherries earn 8 coins. Go hog wild.

Prizes

TM23 Iron Tail: 3500 Coins
One of the powerful Steel-type moves. It can cause great damage and can lower the target's physical Defense allowing for it to be a good coverage move

TM15 Hyper Beam: 5500 Coins
A very powerful Normal-type move. 150 Base Power, but forces you to recharge after using if you don't faint the enemy. Attach it to a strong and fast Pokemon with great Attack power (like Gyarados) and go hog wild.

TM50 Substitute: 7700 Coins
A support move which creates a miniature doll of your Pokemon using a bit of its health to absorb stat-drops, conditions, and damage until its threshold is reached and it breaks. Not as useful as in later generations but can still be used for a good strategy or two.

#030 Nidorina/ #033 Nidorino: 1200 Coins
If you didn't catch one of the Poison-type Nidorans earlier in the game you can get their evolved forms here. Both save you a bit of time training and are pretty good when you evolve them. If you forgot to get a Moon Stone, there's one in the Rocket Hideout in the basement or one at Cinnabar Island.

#035 Clefairy: 500 Coins (Red) 750 Coins (Blue)
Again, this can make up for not getting one of these rare little buggers in Mt. Moon. It too evolves with a Moon Stone and learns a few good Special moves. Normal-type.

#063 Abra: 120 Coins (Blue) 180 Coins (Red)
Makes catching this little Psychic-type a heck of a lot easier now that he can't escape with Teleport. Evolves into a Pokemon with one of the best Special stats in the game (albeit reaching his last form via trade).

#123 Scyther: 5500 Coins (Red)
A formidable Bug/Flying-type Pokemon with great Speed and Attack. Only found in the Red Version and in the Safari Zone, he's much easier to get with slots than randomly tossing balls at him and hoping for the best.

#127 Pinsir: 2500 Coins (Blue)
Another Bug-type Pokemon with a killer stat set version exclusive, this time Pokemon Blue. Again, only found in the Safari Zone for the rest of the game.

#139 Porygon: 6500 Coins (Blue) 9999 Coins (Red)
Exclusive to the slots, Porygon is very expensive in both versions. If you're looking for a strong battler, it's not going to be Porygon, though it has a fun novelty to it you could try and make work with its strong movepool. Normal-type.

#147 Dratini: 2800 Coins (Red) 4600 Coins (Blue)
The only Dragon-type family in the game is found in its genesis only here and at the Safari Zone. Extremely rare and learns a myriad of great attacks, Dratini would make a smooth addition to any team.