How do you organize the files created in Inventor?

First and foremost, do you know what a project file is? And have you created one for your design files. This will be a master workspace for your design files and controls how your templates, styles, and standard content center file are located.

After that it is just a matter of organization that makes sense to the type of work you do. Are you a product based company or a project based company? Do you already have naming scheme for your files? Are you doing this at a company or for personal use?

It depends on the size of the project.
When starting from scratch, i just name the components according to their genral form (beam x, plate x, etc).
But, when a proyect steps forward and you can define de main assemblies, we use a file naming system, like this "AB-CD-12-23-567". "AB" are the poyect area characters. STructures, MAchines, etc. "CD" are correlative alphabetical characters. 12 is for parent assemblies. 34 subassemblies. 567 child pieces (sometimes hundredths become sub-sub-assemblies). And the naming of the file, DWG, IPT and IAM become that code. No sub folders, as you can sort the files alphabetically.

The Vault is the way to go

I agree with Nuno. Vault is the way to go. It can be a bit of work to get figured out the first time, but is definitely worth the time involved.

Once you get the Vault going, the organization is incredible. With the new Vault you can even control the revs of your drawings, you can select what people have access to what files. If you create a library for all your purchased parts, everything will be in one place, you dont need to duplicate parts all over the place, all your assemblies will grab the files from the same locations. No more searching for lost files all over the place.