I've got a site using PHP and MySQL. I've got a domain title reserved with a domain title provider. I don't want to make use of the storage suggested with that provider to keep my files, since it is really too costly.

Let us the domain title is : www.domainname.com

So... Basically choose another provider (let us imagine it's free.fr) to keep my site's files, what may I actually do, or what may I search for, to have the ability to have during my url bar :

http://www.domainname.com/onefolder/onegreatfile.php 

rather than :

http://imhere.free.fr/innerfreefrfolder/onefolder/onegreatfile.php 

Basically activate the transparent redirection on my small domain title, the url keeps being :

http://www.domainname.com

Basically pressure the url bar to visit http://www.domainname.com/onefolder/onegreatfile.php

it returns a 404 not found error (The requested URL /onefolder/onegreatfile.php was not found on this server.)

This is an oversimplification but it ought to be enough to enable you to get where you have to go:

You will want to register having a hosting company that enables you to employ your domain title using their server. They will explain things to set your title servers to to ensure that any demands for www.domainname.com can lead to the ip from the server your files are saved on. Setting the title servers for the domain is generally done using a service supplied by your registrar.

The transparent redirection you are mentioning to (some hosting sites refer to it as Sending with Hiding) is actually just your located files exhibiting inside an iframe in your possessed domain title.

What you will need to consider is setting the DNS servers/Nameservers for the domain title to suggest for your located website/Ip.