Mr. Weasley is the father of the Weasley clan; he is also a kind, generous man. He works everyday at the Misuse of Muggle Artifacts Office of the Ministry of Magic. While he doesn't have a high-paying or prestigious job at the Ministry, there is evidence that he is well liked there: as he walks through the Ministry to escort Harry to his hearing, he is greeted by a number of wizards. The problem is that Mr. Weasley is also a well-known supporter of Professor Dumbledore, who Cornelius Fudge fears and resents. So Mr. Weasley never gets promoted at the Ministry, even though he is a dedicated employee.
As a member of the Order of the Phoenix, Mr. Weasley shares guard duty of Harry's prophecy at the Department of Mysteries. The night that Harry dreams from the perspective of Voldemort's snake, it is poor Mr. Weasley who is sitting up at the Ministry of Magic, and it is Mr. Weasley whom Harry feels himself biting through the snake's mind. The fact that Mr. Weasley is the one attacked gives Harry a reason to spend the Christmas break at Number Twelve, Grimmauld Place with the whole Weasley family. It also gives Harry, Ron, Ginny, and Hermione a reason to be at St. Mungo's over Christmas, which is when they witness Neville visiting his parents in the long-term Spell Damage wing of the hospital. So in addition to the emotional turmoil this attack provokes in Harry, there are also some important narrative reasons that Mr. Weasley is the one he dreams of attacking.
Mr. Weasley's wound proves to us once more Mr. Weasley's curiosity about Muggles. He seems to regard Muggles with the same fascination that we feel towards the wizarding world. So, as Mr. Weasley is lying in St. Mungo's, he decides to try some alternative – a.k.a. Muggle – medicine: stitches! Unfortunately, while stitches work great for our injuries, they don't seem to be recommended for magical snake wounds. When Harry, Ron, Ginny, and Hermione hear Mrs. Weasley say in disbelief, "It sounds as though you've been trying to sew your skin back together [...] but even you, Arthur, wouldn't be that stupid —" (23.190), they all rush for the door. Mr. Weasley's curiosity about Muggles adds a touch of humor to a situation that is otherwise deadly serious.
In fact, Mr. Weasley's love of Muggles is yet another reason why he's not advancing at the Ministry: anti-Muggle prejudice interferes with Mr. Weasley's own career. And Mr. Weasley's career troubles are the cause of a major struggle in the Weasley household in Book 5.
Apparently, after Percy Weasley's boss, Barty Crouch, Sr., turned out to be under the Imperius Curse in Book 4, everyone thought Percy (the third oldest Weasley child) would get into trouble at the Ministry for not noticing his boss's enchantment. But instead, Percy got a big promotion. He's now Cornelius Fudge's secretary. When Percy came home to brag about his promotion, Mr. Weasley suggests, "Fudge only wants Percy in his office because he wants to use him to spy on the [Weasley] family – and Dumbledore" (4.153).
Percy can't stand to think that his promotion wasn't the result of his own hard work and ambition. So Percy has broken with the Weasley family and basically disowned his dad. What's more, Percy has told his father that he has "been having to struggle against Dad's lousy reputation ever since he joined the Ministry and that Dad's got no ambition [... and that] Dad was an idiot to run around with Dumbledore" (4.157-159). Percy blames Mr. Weasley and his connections to Dumbledore for the family's poverty. Of course, Mr. Weasley stays loyal to Dumbledore and to the Order. But the fact that he is not on speaking terms with Percy causes the family a lot of anxiety, especially Mrs. Weasley.