We are looking at upgrading our server and the company who would provide it is asking this question:

"Your Adagio software, does it run on SQL, or does it have its own built in database?"

Am I'm correct in that Adagio software has it's own database and it is not SQL. Can anyone confirm?

Adagio has its own built in database and does not require or use any external database (MS SQL or otherwise). However, hardware that makes MS SQL run fast will also make Adagio run fast.

For the server, invest money in fast, high performance, mirrored disk drives.