-How he got it

Because Inutaishou died before Inuyasha was hold enough to use a weapon, and because Sesshoumaru wanted the blade for himself, Tessaiga was hidden inside Inutaishou's grave. The means to get there was the black pearl, which was then hidden inside Inuyasha's right eye. Sesshoumaru discovered the location of the pearl, pulled it out, and used Jaken's staff to open it. Kagome managed to pull the Tessaiga out of the stand it was in and she then handed it over to Inuyasha.

-Uses

--Seal
Tessaiga was originally forged to protect Inuyasha. Because the blood he inherited from his father was too powerful for a hanyou's body, the blade sealed. Otherwise, the blood would take over him when he is in a threatening situation in order to protect him. Unfortunately, the blade was broken in a battle with Goshinki and the seal was broken as a result. Since then the blade has been reforged and the seal only works when Inuyasha is holding onto the Tessaiga.

--Scabbard
Both Tessaiga and Tenseiga's scabbards came from the wood of an old tree named Bokusenou and are strong enough to withstand lightning attacks (two examples being Hiten and Ryukossei). The scabbard is also hard enough to block weapons. He uses it against Hiten to block his Raigekijin, and uses it again later when Sesshoumaru wields Tessaiga with a human arm.

--Barrier
The barrier of Tessaiga keeps youkai from holding or using it. There have been a few exceptions - Toutousai can hold it (which would make sense, as he was the one who made it) and Shippou carried it in a filler anime episode. It was also used during the fight against the leader of the spider-heads to destroy his silk and block him from entering the alcove they were hiding in.

--Cutting
This is the method Inuyasha used until chapter 128, before he learned how the Kaze no Kizu worked. Though he can't kill multiple youkai at once while using this technique, it is still fairly efficient. What's more, when Tessaiga cuts through a strong opponent it absorbs their powers.

--Kaze no Kizu
--(Scar of the Wind)
This is the basic power of Tessaiga. Inuyasha first sees the Kaze no Kizu demonstrated in chapter 60, when Sesshoumaru uses a human arm to steal the Tessaiga. He later uses it by accident in chapter 105 while protecting Miroku. It isn't until Toutousai makes his appearance that Inuyasha learns how to use it: by striking through the space where the winds collide. Later, in the battle against Ryukossei, the Kaze no Kizu wraps around the blade so that Inuyasha can use it whenever he wants.

--Bakuryuuha
--(Backlash Wave)
The Bakuryuuha is Tessaiga's ultimate technique, and Inuyasha first successfully performs it while he is fighting with Ryukossei. It uses the Kaze no Kizu as a sort of starting point: by hitting the enemy's attack in the right spot with the Kaze no Kizu, it turns the enemy's own demonic power against them. The demonic power and the Kaze no Kizu join and create giant twisters that rip an opponent apart. The trick to using the ability is that the one attempting it has to have a more powerful demon aura than their enemy's.

--Red Tessaiga
After facing one of Naraku's barriers, Inuyasha decides he needs to find a way to make the Tessaiga stronger. Myouga tells him about a family of Hyakki bats in the west that are protected by an unbeatable barrier. Each generation has a protector, and if Inuyasha manages to kill the protector he will be able to break through barriers. Unfortunately, the protector of the barrier turns out to be a hanyou girl named Shiori. Inuyasha kills her grandfather and saves her village, and she lets him cut through a red pearl. The jewel has been handed down through the generations and uses the keeper's power to create the barrier. After a few unsuccessful attempts, Inuyasha manages to cut through the pearl. The blade of Tessaiga turns red and he gains the ability to break through barriers.

--Kongousouha
On the second trip to the youkai graveyard, the Inuyasha-tachi and Sesshoumaru are fighting against Naraku for the final shard which Housenki has taken so the jewel cannot be completed. Because Inuyasha is more concerned with saving his friends then gaining a new technique for Tessaiga, Housenki lets him inherit his youkai power. Tessaiga becomes encased in diamonds, which can then be shot at the opponent.