First lets be clear that this is involving the books, the movies don't explain/show some things differently and certainly leave a lot more plot and background holes than the books (although the books leave plenty). Also lets be clear that work is extremely slow for me today so I can put quite a bit of thought into this...
Now, how exactly does the economy work in the world of Harry Potter? It seems to me with magic there would be little need for money, you can conjure up most of the things you need (shelter, clothing, entertainment), you can't create food but you can increase the quantity if you have some food, and you have little to no transportation costs bc you can apparate and take the Flu Network. Also, although the wizard world does not use muggle money, it seems to me that if you knew magic you could easily produce or do things to make a lot of money in the muggle world, and use it to buy things like food, housing, and other essentials. Honestly it's shocking how little wizards seem to know about muggle society or muggle money, why wouldn't they get high paying jobs and use that money to fill their needs?
Also there is the less talked about use of house elves, which are free laborers, so basically they are slaves and most of them are happy about it. All you need is a few house elves to do all your housekeeping or farming for you, hell you could probably send a house elf out to get a job and make you money.
Now you need money for things like school supplies, wands, going out to eat/bars, magical toys like broomsticks, ect... But when you don't need money for so many other things it should be pretty easy to keep a little spending money around I would think.
All of that being said money is clearly a big deal in the wizard world, just look at how the Weasley's are ridiculed by wealthier families like the Malfoy's, and how big of a deal it was that Harry's parents left him all that money, and how much power the goblins at Gringotts have. Even still though the Weasley's can't be hurting THAT badly since they can afford to bring the whole family to the Quidditch World Cup, and 3 of the sons seem to have legitimate jobs that would pay (not to mention Arthur has a stable job).
So it is kind of confusing that they care so much about money, since it seems to me they really don't have a need for it. My best personal explanation is that having name brand/new/expensive things is more important in the wizarding world than the muggle world. You don't need money but having it gives you some sort of superiority over others. Still though I don't see why the Weasley's couldn't have magiced themselves nicer things and a better house, why didn't they?
Here is an explanation I found online
A part of this is more or less explained in "Quidditch through the ages." It's simple really. Different wizards and witches have different talents and/or actual powers. Not everybody can make a broomstick fly or charm a dangerous beast. As such, different abilities and attributes give certain people obvious jobs. And as such, you need currency. Sure, most wizards and witches could survive by themselves, conjuring up food and such, but it'd be less then the ideal life. With the exception of ghosts and immortals, few could live long enough to learn everything about magic and become completely self sufficient. They have magic, not omnipotence.
The simple answer would be, they can't create anything they want. They do not live in boundary-less world. They can't just conjure money out of nowhere. Well, they probably can. But the Ministry of Magic probably keeps that illegal, for obvious reasons. It would be like counterfeiting.
It's never clearly stated, but I suspect that they can't just create anything they want. It still has to be made from existing raw materials, even though magic replaces technology in assisting the production process. Note that there aren't any Hogwarts classes in creation: transfiguration, illusion, summoning, transporting, yes, but not creating something from scratch that didn't exist before. If you did manage to create something by magic, it would be made of magic, and probably not have quite the same properties as even an enchanted physical object would.
Even if I'm wrong about that, the time-honored reason to have a medium of exchange still holds true: Specialization. Either there are some things that only some people know how to make, or some people make some things better than others. You yourself could go to the fabric store and sew your own clothes--assuming you can sew--heck, you could grow the cotton and weave and dye it, but it's easier to go to a store and buy clothing ready-made. Given that there are brand names for things, the wizarding world has both specialization and mass production.
I guess this makes sense but still the Weasley's have proven to be excellent wizards so I don't see where magic skills play a part in it. I don't know the answer to be honest, why is money such a big deal to wizards when they can live actually very successful lives without it?
Wow this post is long, I guess I will talk about the wizarding government (which seems like an Oligarchy more so than a Democracy even in the earlier books) later on.