The will of Sirius Black left all of the House of Black's and Sirius's personal possessions to the owner's godson, Harry Potter.
Sirius's Possessions
Kreacher
When Harry realized he now owned the house-elf that was responsible for his godfather's death, he wanted
Kreacher
nothing to do with him. To compromise, Albus Dumbledore suggested that he [Harry] leave Kreacher at Hogwarts School of Witchcraft and Wizardry. It wasn't until Harry needed Kreacher during the Horcrux hunt and Kreacher became more likable that Harry began to care about the elderly house-elf.
12, Grimmauld Place
After Dumbledore told Harry that Sirius left Harry 12, Grimmauld Place, Harry said the Order of the Phoenix could continue running it because he wanted nothing to do with it. He, also, said specifically that he didn't want
12, Grimmauld Place
Sirius's killer Bellatrix Lestrange to have it. The Order stopped using 12, Grimmauld Place after it was thought that Severus Snape was a double agent and went to use The Burrow. When that happened, Alastor "Mad-Eye" Moody set up jinxes to stop Snape.
Later, after the Skirmish on Tottenham Court Road, Harry and his friends Ronald Weasley and Hermione Granger hid out in Grimmauld Place, while searching for Horcruxes. It wasn't until, after the Infiltration of the Ministry of Magic, that they had to leave #12 because they accidentally gave away their position to Yaxley.
R.A.B.'s Locket
One possession of Harry's played a big part in turning Kreacher on his side - the locket of R.A.B. Harry didn't find out
R.A.B.'s locket
that he owned it until 1997 when he learned that R.A.B. was a member of the Black family - Regulus. After asking Kreacher to tell the story of how Regulus died, he gave the locket to the elderly house-elf.
Mundungus Fletcher
Harry came into feuds with the sneak thief Mundungus Fletcher over the Sirius's possessions. One occurred in 1996, where Harry attacked
Mundungus Fletcher
Mundungus after seeing him selling Sirius's two-way mirror to Aberforth Dumbledore. Another one occurred in 1997, during the Horcrux hunt, where he asked Mundungus the location of
The Marauders photograph
Salazar Slytherin's locket. He revealed that the location was with Dolores Umbridge.
Lily Evans, author of said letter
The Marauders
In 1997, Harry found another possession of his in Sirius Black's bedroom - a photograph of the Marauders (James Potter I,Sirius, Peter Pettigrew, and Remus Lupin. Unfortunately, a Permanent Sticking Charm prevented him from taking it.
Lily Potter I's Letter
Also, in Sirius's room, Harry found a letter written by his mother to Sirius. Since it was Sirius's, it now belonged to Harry, so Harry took it with, along with a photo of him with his his toy broomstick. All his now since Sirius's death.