Thanks everyone for the responses. I'm a hobbyist programmer so my self-training has many holes. CheckDB was one of those holes. However, running it now seemed relatively positive, as much as I can make out the output. It was much like the sample output on the MSDN page. And at the end it said,

CHECKDB found 0 allocation errors and 0 consistency errors in database 'mydatabase'

So that was a relief of sorts.

However, I noticed something that should have been obvious to me the first time around that is perhaps a related detail.

In the database listing in SQL Management Studio it has (Principal, Suspended) behind the database name. This is on the web host and after a quick search it seems that this is related to a database mirroring arrangement (that I didn't know about). Could this cause the backup problem?