The 10 votes remaining means you have 10 remaining delete votes.

It may be documented somewhere, but we can reasonably assume that's how it works, since the Close Vote dialog mentions *50 votes remaining* which is [known](http://meta.stackoverflow.com/questions/193049/meaning-of-votes-remaining-in-close-vote-dialog) to be your own limit, and the flag dialog mentions *23 flags remaining* which is also your own limit. (Numbers may vary.)

**Edit**
Here's the feature request for the implementation of this: http://meta.stackoverflow.com/questions/204428/can-we-display-the-users-remaining-votes-when-voting-to-delete