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Alastor "Mad-Eye" Moody, or simply Mad-Eye, introduced during The Goblet of Fire book, is a powerful pure-blood Auror. However, later in his career as he lost one of his eyes, and replaced it with a magical one, people seem to have started calling him simply Mad-Eye.
Did Alastor know of that nickname people came up with for him?
Yes. Tonks calls him it directly at least twice in The Advance Guard, OotP.
"Nice one, Mad-Eye," said Tonks appreciatively
"We're not going through clouds!" shouted Tonks angrily, "we'll get soaked, Mad-Eye!"
We know they're close, and she's his protégée at the Ministry, but I doubt he thinks it's only her nickname for him. She also uses it around other people, which indicates it isn't just between the two of them, and others are expected to know it.
Sirius also calls him that directly at least once too.
Sirius had just said, "What's that you've got there, Mad-Eye?" and Moody had turned towards him.
The Woes of Mrs Weasley, OotP
In GoF, although he isn't actually himself for it (so the real Moody wouldn't know from this source), Rita Skeeter refers to him as 'Mad-Eye' in one of her articles.
"In September of this year, he hired Alastor "Mad-Eye" Moody, the notoriously jinx-happy ex-Auror... Mad-Eye Moody, however, looks responsible and kindly"
Dumbledore's Giant Mistake, Daily Prophet
Padfoot Returns, GoF
She is often rude about the subjects of her articles, but she is using the name as though it's common knowledge. As The Daily Prophet is the wizarding newspaper with the largest circulation, it wouldn't be surprising that Moody would read it.
Yes, Tonks calls him it to his face several times throughout the books, one example below.
“We’re not discussing anything here, it’s too risky,” said Moody, turning his normal eye on Harry; his magical eye remained pointing up at the ceiling. “Damn it,” he added angrily, putting a hand up to the magical eye, “it keeps sticking — ever since that scum wore it —”
And with a nasty squelching sound much like a plunger being pulled from a sink, he popped out his eye.
“Mad-Eye, you do know that’s disgusting, don’t you?” said Tonks conversationally.
Harry Potter and the Order of the Phoenix
Later on we see Sirius also call him Mad-Eye, so we know that Alastor is aware it's relatively common amongst those who know him and it's not just limited to Tonks.
He was spared the trouble of inventing an object he had not packed; Sirius had just said, “What’s that you’ve got there, Mad-Eye?” and Moody had turned toward him. Harry crossed the kitchen, slipped through the door and up the stairs before anyone could call him back.
Harry Potter and the Order of the Phoenix
I could probably gather more quotes of people doing so but these quotes should suffice in answering the question. Even if no one would say it to his face I doubt he would be completely unaware of the nickname anyway.
