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Throughout Book 4, we got to know a man who looked and sounded like Alastor Moody, but in fact, he was a Death Eater using Polyjuice Potion to disguise himself as Moody. In Book 5, we get to see the real Moody: he is as gruff as the fake Moody but perhaps even more paranoid, which makes sense given his experiences in the previous Harry Potter novel.
Moody is the head of the group of Order members sent to escort Harry from Little Whinging to Number Twelve, Grimmauld Place. He is a dedicated Dark wizard fighter, even if he is a little eccentric looking: he has a revolving, magical glass eye (hence the nickname "Mad-Eye") and he is also missing a chunk of his nose. He checks and double checks all of their flight plans and includes about a dozen fail-safes in case they are attack from the ground or by air (none of which happens, of course). Real Moody's motto still seems to be "Constant vigilance!", much as fake Moody used to say in Book 4.
When Mrs. Weasley holds a party to celebrate Ron and Hermione being appointed prefects, Moody comes bearing a photo album of the first round of Order of the Phoenix members. He shows them to Harry during the party. Harry feels ambushed by this photo album, which shows a whole generation of young, optimistic wizards and witches, Harry's parents among them.
Now, look at what has happened to the original order: James and Lily Potter have been murdered, Frank and Alice Longbottom have been tortured to insanity, and on and on. Moody means well, but he is not at all emotionally sensitive, and showing Harry this photograph leaves him feeling even more shaken and upset than he was already (if that is imaginable, given the emotional turmoil with which Harry starts Order of the Phoenix).
Moody also ends the novel by threatening Harry's Uncle Vernon with unspecified vengeance if the Dursleys don't treat Harry right over the summer. Given Moody's terrifying appearance and magical eye, we're pretty sure that we would be intimidated if he threatened us. So we think he is a good choice to terrify the Dursleys into obedience.