What happened to the Orcs after the Ring was destroyed and Sauron was defeated?

The Orcs were actually creatures of Morgoth, not of Sauron, being corrupted by him during the years of the Trees.

Since Morgoth dissippated his power into Arda, and since he was removed from the world at the end of the First Age, the defeat of Sauron has no effect on them so long as general survivability is concerned.

Also, there have been extended periods of time between the defeat of Morgoth and the final defeat of Sauron when Sauron was not active and could not command them. These times were as recent as TA 2950 (i.e before Sauron declared himself openly again) and were what Shagrat and Gorbag refer to in their discussion in the Two Towers:

'They would,' grunted Gorbag. 'We'll see. But anyway, if it does go well, there should be a lot more room. What d'you say? – if we get a chance, you and me'll slip off and set up somewhere on our own with a few trusty lads, somewhere where there's good loot nice and handy, and no big bosses.'
'Ah!' said Shagrat. 'Like old times.'

There's no reason to suppose that post-Sauron times would be any different from these "old times". In other words, the Orcs would resort to independent banditry rather than being an organized force.

Some of the orcs in Mordor were bred by Sauron:

...a race of sentient beings bred by the evil Vala Melkor (Morgoth) during the time of the Great Darkness. The Dark Lord Sauron also bred them, and later the wizard Saruman

It is stated that:

After the ultimate defeat of Sauron, Mordor became mostly empty again as the orcs inside it fled or were killed. Crippled by thousands of years of abuse and neglect, but capable of sustaining life, the land of Mordor was given to the defeated foes of Gondor as a consolation

Assuming that they lived and bred as Elves and Men do, the rest that fled would have gone "home". Those that Sauron didn't create would have probably gone to Angmar and Ered Mithrin.

No female orcs are ever mentioned by Tolkien, but in The Silmarillion he wrote that "the Orcs had life and multiplied after the manner of the Children of Ilúvatar (Elves and Men)"

They were nearly destroyed in the War of Wrath, and those that survived fled eastwards into the Mountains of Angmar and the Grey Mountains (Ered Mithrin).

We may assume that the others (the ones Sauron created), along with some of the Melkor breeds, fled and hid anywhere they could. Inside deep caves, pits or chambers of fallen fortresses, and of course the Misty Mountains (Hithaeglir).

Although the entire force of Sauron was extinguished by the end of the War of the Ring, groups of orcs were thought to continue to dwell in the Misty Mountains, but posed no threat to the lands north, south, east and west after that point.

Almost all of the orc army force was destroyed, leaving them crippled as a race since we don't know anything about the dwellings of female orcs and child orcs.

Orcs continue to live, hiding, until our days. Since the fall of Sauron, they lacked the strong will that was driving them, and they scattered and their numbers decreased. Also, as while Sauron was in the Middle Earth there was a shadow on the heart of free people (that led them to fear, despair or corruption), after the Dark Lord demise, all the dark creatures began to have a similar weight.

If we take the Hobbit as accurate (it isn't in some matters that changed later), orcs did not only survive, but they are responsible for many of the harmful inventions (probably bombs and other weapons).

I think the orcs made home in Dol Goldur or Angmar. But the silmalirion says that the orcs of Angband fled to the grey mountains.

It is only known that the slave Orcs of Mordor were set free. The fate of the Orcs in the Misty Mountains is not known.

Gandalf stated that he pitied the slaves of Mordor:

'You think, as is your wont, my lord, of Gondor only,' said Gandalf. 'Yet there are other men and other lives, and time still to be. And for me, I pity even his slaves.'

Lord of the Rings: The Return of the King - The Siege of Gondor

No doubt this view was shared by Aragorn also:

... and the slaves of Mordor he released and gave to them all the lands about Lake Nurnen to be their own.

Lord of the Rings: The Return of the King - The Steward and the King

For the Orcs that were still rebellious and attacked the Kingdoms of Men, they were slain:

For though Sauron had passed, the hatreds and evils that he bred had not died, and the King of the West had many enemies to subdue before the White Tree could grow in peace; ...

Lord of the Rings: The Return of the King - Appendix A, Of Rohan

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The fate of the Orcs in the Misty Mountains is not known, but presumably they stayed there and lived in independence. They were most likely slain if they attacked.

The squint-eyed Southerner at Bree who was a companion of Bill Ferny was remarked by Merry to look as orc-like as some of Saruman's soldiers marching away in Flotsam and Jetsam. Likely some survived the Wrath and lived as was their wont. Shagrat was probably right.

If orcs are corrupted Elves, then they eventually faded as did the Elves who remained in Middle Earth.

If orcs are corrupted Men, then the most corrupt ones probably got killed in whatever wars and banditry they perpetuated, and the race as a whole mellowed out over time.