This is the Al Bhed salvage ship. When Tidus regains consciousness, you'll find him surrounded by people who don't seem to speak a word of English. That is, until your saviour does.
The Salvage Ship
When you get control of Tidus, talk to the man on the left behind the guy who walked out with the girl. He'll give you 2 Hi-Potions. Go and save if you really want to, although it doesn't particularly matter. Talk to the girl once you're finished, and you'll be treated to a nice tutorial on how to use the Sphere Grid, this game's levelling-up system. It's important, but if you miss it or forget how to use it, you can view it again from any help console scattered throughout the rest of the game, or here, located elsewhere in the guide.
Once that's finished, make sure you upgrade Tidus, then talk to the girl again, and she'll explain what you have to do. The on-screen controls will tell you how to navigate in water, so look at your map and head towards the red blinking arrow to your destination. Other than looking at the map, you can also follow the chain down to the sunken ruins. On the way, you may run into some random battles. None of them are particularly difficult, mostly against Pirahnas which are easy to kill. Note that the girl in your party takes far too much AP to level up than you would reasonably be able to acquire here, so don't worry about it. If you want to participate in a few battles to level up Tidus a bit, then you can.
Underwater Ruins
Once you've followed the chain to the entrance of the ruins, you'll go inside. Use the Save Sphere if you want, then head up and examine the console. The doors will open, and you'll head inside. Follow the girl along the tunnel to the room at the end, where you'll get into an unavoidable fight with two groups of Pirahnas. Make sure you Steal some Grenades from them, since you'll need them for an upcoming battle. Otherwise two hits from Tidus or one from him and one from the girl will kill a group of Pirahnas.
After the battle, examine the machine at the top of this room (the girl will be pointing at it) and you'll get a scene with the ruins powering up, and a monster closing in on you. You'll swim out, and be confronted by...
BOSS: Tros
It's a giant squid! Use Cheer with Tidus if you want to help him and the girl out a little, otherwise attack it with Tidus and use the girl to throw Grenades. A hit from Tidus will do about 150 damage, and a Grenade will do about 350. Steal more Grenades from Tros if you run out. The Trigger Commands in this battle are self-explanatory, and if Tros decides to swim away from you make sure you close the gap again with Pincer Attack (under Tidus' commands) or he'll use Nautilus Charge (you can't avoid this the first time) that will do about 170 damage to both. His ordinary tentacle attack will do 50-60 damage to one person, but it'll miss often. After 5 or 6 Grenades and a couple of attacks from Tidus, Tros will die.
Afterwards, swim towards the hole where the girl is. Follow the beam of light and the red arrow down to where the girl is waiting further towards the sea bed, to a wreck of an airship. After the scene, you'll be back on the deck of the salvage ship. After talking a bit and a few more scenes, you'll fall into the ocean, and all will go black.