Tessaiga
(Tetsusaiga)

-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.

Tenseiga

-How he got it

--Teleportation
When Sesshoumaru was hit by the Kaze no Kizu in a battle with Inuyasha, Tenseiga surrounded him with a black light to protect him from the blast and transported his body out of harm's way. It seems to be a built in defense mechanism, because it acted of it's own accord.

--Communication
Tenseiga pulsates and/or rattles in it's sheath frequently when it wants Sesshoumaru to pay attention to it. Generally it wants him to go see something or be used, which Sesshoumaru does.

--Resurrection
Tenseiga's obvious use. Though the blade cannot cut anything alive, it can be used to cut through the messengers from the other world that come to take the souls of the deceased. When these figures are 'killed' in a sense, the person they came for comes back to life. The Tenseiga has limits, however - it can only be used on someone that is actually dead, and there has to be a body.

--Souryuuha
--(Movie only)
This attack was created for the third movie. A blast of light/energy is sent at the target. Souryuuha is powerful enough to deflect Tessaiga's Bakuryuuha, which Sesshoumaru combines it with at the end of the movie to defeat Sou'unga.

--‘100 with one swing’
Though it has not actually been shown, Toutousai told the Inuyasha-tachi that Tenseiga has the ability to save a hundred souls with one stroke, which would counterbalance Tessaiga's power to destroy a hundred with one stroke.

Both

As Toutousai put it, the swords are equals, each no less or no greater than the other (though Sesshoumaru feels differently). They serve different purposes, one to kill and and one to save, but they were both designed to protect. As a hanyou, Inuyasha had more need for Tessaiga's protection, both to keep his body from harm and to stop his demon blood from taking over his mind. Tenseiga protects Sesshoumaru from dying, and seems to be teaching him compassion. The two swords also keep the brothers from killing one another. Tenseiga can't be used in battle and it protects Sesshoumaru from Tessaiga's force.

Also, with the Toukijin fight, Goshinki's desire still fueled the blade. This has made me wonder if Inutaishou's will is still living on in Tessaiga and Tenseiga in some why or another, since Tenseiga has a will of it’s own. It’s a nice thought, since the swords have saved Sesshoumaru and Inuyasha's lives, here:

And here: