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 Meowth, Koffing, or Arbok win 15 coins, and three cherries earn 8 coins. Go hog wild.
Prizes
TMs
TM23 Dragon Rage: 3300 Coins
A decent move and the only Dragon-type one in the game. Will always deal 40HP damage to anything regardless of type, stats, or weakness. Can be learned by the Dragons and dragonesque Pokemon including Charizard and Aerodactyl. You can probably figure out better options for these Pokemon later in the game but if you have some money to blow after the fourth badge it can be helpful to have a set damage attack.
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 nuts.
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.
Pokemon
#037 Vulpix: 1000 Coins
Vulpix the Fire-type fox is exclusive to the Game Corner in the Yellow version of Pokemon. Ninetales, his evolutionary form, can only be obtained with a Fire Stone purchased at Celadon Dept. Store. An expensive though balanced fox.
#040 Wigglytuff: 2680 Coins
Wigglytuff are available on demand here for just 2680 coins. If you don't want to use a Moon Stone on Jigglypuff, this is your only alternative other than to trade one in.
#063 Abra: 230 Coins
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: 6500 Coins
A formidable Bug/Flying-type Pokemon with great Speed and Attack. Scyther can only be found here and in the Safari Zone, and he's much easier to get with slots than randomly tossing balls at him hoping for the best.
#127 Pinsir: 6500 Coins
Another Bug-type Pokemon with a killer stat set that is only otherwise found in the Safari Zone.
#139 Porygon: 9999 Coins
Exclusive to the slots, Porygon is very expensive. 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.