Images can be tricky. If you just want an application that will work, is free and is already written for php and mysql look into Gallery2. Pretty easy to install and gives you options for customer logins.
Just to clarify Gallery2 IS a pre-built web application that uses PHP and MySQL to store your images on the server. It creates thumbnails, allows for restricting access to specific users and is easy to administer and install.
The most important thing is to have something you are comfortable using and personally I try to pick an application that has been out for a while. Some new applications can be easily hacked or are know to be front ends for hackers to get access to your server. If you find a package you like the look of Google it and add "hacks" to the search (i.e. "gallery2 hacks") and see what pops up. It is better to be safe than sorry.
I understand and again I suggest gallery2 It is easy to install and the basic setup will do what you want. Should you choose to make it more complex in the future the options are there. You unpack the package and a basic setup takes about 1 minute and you can start posting images. No compiling, no configuration other than having a database in mysql for it to use, which it creates for you. Check out CNET for suggestions on different packages for different operations.