From Bulbapedia, the community-driven Pokémon encyclopedia.
The S.S. Anne, or the St. Anne as it is called in the anime, is a luxury cruise liner which sails the world, stopping annually in Vermilion City.
In the Games
To get on the S.S. Anne in the games, you must get an S.S. Ticket after helping out Bill.
This ship contains many Pokémon trainers looking for a Pokémon battle, including many Gentlemen and Sailors. Battling through the ship is easy, however, because a room is provided with a bed that acts like a Pokémon Center, restoring all Pokémon onhand to full health. Before reaching the Captain's Cabin, you will encounter your rival and battle him.
Afterwards, in the Captain's Cabin, you will find that the Captain is seasick. If you help him out by giving him a back rub, he will provide you with the HM for Cut, which can be used, among other things, to gain access to the Vermilion Gym.
Once you have the HM001, the S.S. Anne will sail away the moment you step off the ship, never to be seen again.
In Pokémon Red and Blue, there is a truck which appears in the S.S. Anne's port which serves no evident purpose except as scenery. With scenery being very sparse in the Generation I games, and this being one of the few sprites with no other purpose than to exist in one spot, fans began speculating that another purpose existed for this truck, and that it could perhaps be moved with Strength. Thus, a popular rumor sprang up that a Poké Ball containing Mew could be found underneath. The only flaw with this was that the HM for Strength could only be obtained after getting Cut, which required the S.S. Anne leaving. Players found ways around this by trading Pokémon knowing Cut to other games which did not have the HM.
The rumor was proven false.
In the Anime
The St. Anne appeared in Battle Aboard the St. Anne and Pokémon Shipwreck.
Ash, Misty and Brock boarded the ship after two schoolgirls, who were actually Jessie and James in disguise, offered them free tickets. This was part of a Team Rocket plot to steal the Pokémon of all trainers onboard. However, Ash and his friends enjoyed the St. Anne, which provided large amounts of food for people onboard and allowed Pokémon battles belowdeck. While onboard, James was swindled by the Magikarp Salesman into buying a Magikarp in a gold-plated Pokéball, and Ash traded his Butterfree to a Gentleman for a Raticate.
Out on the open ocean, Team Rocket, including Jessie, James, Meowth, and a number of Team Rocket Grunts began taking the Pokémon of everyone else onboard the St. Anne by force. Ash led a counterattack against the Rockets, leading to a large battle and many of the members of Team Rocket being 'blasted off' through the side of the ship.
Afterwards, the St. Anne started taking on water, and the passengers had to be evacuated. Ash delayed evacuating to trade back the Raticate for his Butterfree, and while rushing to escape the St. Anne jerked and knocked out Ash, his friends, and the Rocket trio. When they awoke, the St. Anne was upside down and underwater, balancing on a seamount.
The two groups fought at first, but Misty took charge and began directing all of them in an emergency escape plan before the ship fell into the depths of the sea floor. Since the ship was upside down, they would need to escape through the thinnest point in its hull, which Misty knew was in the engine room, having built a model of the ship herself. They made their way through the ship to this point, and then Ash ordered his Charmander to burn a hole in the hull. With the help of Squirtle, Goldeen and Starmie, Ash and his friends were able to swim to safety, while Team Rocket nearly drowned since all they had to rely on was James's Magikarp, which could only Splash around on the floor.
Once everyone was offboard, the St. Anne's balance tilted, and it fell off the seamount, down towards the abyss, never to be seen again.