Photo Galleries - photos on our web host - options

Our current photo gallery is Coppermine thru our web site/host. I think this is a SQL database and operated thru the host (per say). We have ADMIN rights to setup folders, add photos, and such. This has been more problematic than its worth. Seems every time I want to use it, there are issues and I'm contacting tech support to get it fixed. It's time for something new and a complete overhaul.

I don't even know 'what' to ask...or how this all works.

We need a new photo gallery. I don't even know what my options are - but, we want to utilize our web space. Photos must be hosted on our website. In other words...no photobucket or nothing that you upload photos to an unknown location.

I don't know if its a new SQL or other software (????) that gets put on our web host (if they support it) ?

Or if its an outside website that will access your web host space - but allows you to organize photos from another link location?? I have no frigin idea....

So what options are there for photo galleries?
Ones that I can host the photos on my own web site space....

Or any thoughts?