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In Chamber of Secrets he tries to wipe Harry and Ron's memories with Obliviate (Memory Charm) using Ron's damaged wand.

It backfires, and he loses his memory. What happened to him next?

Harry and Ron run into Lockhart at St.Mungo's, while they were there to visit Arthur Weasley after the Nagini attack in the Order of the Phoenix.

A man was peering out at them all with his nose pressed against the
glass. He had wavy blond hair, bright blue eyes, and a broad vacant
smile that revealed dazzlingly white teeth.

“Blimey!” said Ron, also staring at the man.

“Oh my goodness,” said Hermione suddenly, sounding breathless.

“Professor Lockhart!” - Harry Potter & The Order of The Phoenix, Ch.23 - Christmas on the closed ward.

According to the nurse, he was slowly improving, but he still doesn't remember a thing. He instinctively writes autographs to everybody. The healer responsible also mentions that nobody visits him, and is happy to see that he has visitors (Mistaking Harry, Ron & Hermione for people visiting him).

A motherly looking Healer wearing a tinsel wreath in her hair came
bustling up the corridor, smiling warmly at Harry and the others. “Oh
Gilderoy, you’ve got visitors! How lovely,and on Christmas Day too!
Do you know, he never gets visitors, poor lamb, and I can’t think why,
he’s such a sweetie, aren’t you?"

In Order of the Phoenix, there’s a visit to St. Mungo’s Hospital in which they meet Lockhart again. He’s still recovering from the effects of the memory charm, but their are signs that his flamboyance and love of fame are recovering:

“I’m very well indeed, thank you!” said Lockhart exuberantly, pulling a rather battered peacockfeather quill from his pocket. “Now, how many autographs would you like? I can do joined-up writing now, you know!”

But that’s three years after the incident in the Chamber of Secrets, and according to the nurse on the ward, this seems like quite a recent development:

“Listen to him,” said the Healer, taking Lockhart’s arm and beaming fondly at him as though he were a precocious two-year-old. “He was rather well known a few years ago; we very much hope that this liking for giving autographs is a sign that his memory might be starting to come back. Will you step this way? He’s in a closed ward, you know, he must have slipped out while I was bringing in the Christmas presents, the door’s usually kept locked… not that he’s dangerous! But,” she lowered her voice to a whisper, “he’s a bit of a danger to himself, bless him… doesn’t know who he is, you see, wanders off and can’t remember how to get back… it is nice of you to have come to see him.”

Indeed, his nurse describes his condition as permanent:

“This is our long-term residents’ ward,” she informed Harry, Ron, Hermione and Ginny in a low voice. “For permanent spell damage, you know.”

He’s also still getting fan mail, but he has no idea why. He continues to sign and write to his fans, presumably just because it’s something he seems to enjoy. Residual memories, etc.

“I am not forgotten, you know, no, I still receive a very great deal of fan mail... Gladys Gudgeon writes weekly... I just wish I knew why.” He paused, looking faintly puzzled, then beamed again and returned to his signing with renewed vigour.

Although none of these people care to visit the man in person. According to his nurse:

“Oh, Gilderoy, you’ve got visitors! How lovely, and on Christmas Day, too! Do you know, he never gets visitors, poor lamb, and I can’t think why, he’s such a sweetie, aren’t you?”

Finally, JK Rowling confirmed in a Bloomsbury web chat in 2007 that he probably was stuck on this ward for life, despite some signs of minor recovery:

Jessie: Will Lockhart ever recover?

J.K. Rowling: No. Nor would I want him to. He’s happy where he is, and I’m happier without him!

Finally, it should be noted that this is only what happens in the books. In the films, there’s a short post-credits scene to Chamber of Secrets which zooms in on the window of Flourish and Blotts to show Lockhart on the cover of a book Who Am I?, which depicts him in a strait jacket and mumbling incoherently. No details are given about whether he wrote the book himself, or whether it was written by another person about him.

Somebody has uploaded this scene to YouTube: