After the installation I copied all the files from the test server including database backups. And when I try to restore, I get the following:

Zenphoto encountered an error
MySQL Query ( SHOW FULL COLUMNS FROM [prefix]admintoobject ) failed. MySQL returned the error Table 'tricky_zen._admintoobject' doesn't exist

Here is the list of possible problems:

1. You have not restored the database on the new install. It is not enough to copy the backups, they have to be restored on the new database.

2. The "prefix" for the tables has changed. Be sure that you have told setup to use the same prefix as from the test server.

3. You have not run setup on the new server. You must always run setup if you move an installaton.

Seems that the problem is table prefixes. How can I edit that?

I mean, perhaps, there is a file where this settings is written down?

zp-data/zenphoto.cfg. You also have been asked for it when running setup the first time.