The magic names are Latin and they are chosen to represent the wishes of the magic user. The numbers at the end serve as unique identifiers in the case that the term has been used previously as a magic name.
This is a quote from Tsuchimikado from page 112 of volume 4 of the Yen Press translation of A Certain Magical Index:
"Sorcerers-particularly the kind who assumed a foothold in the nineteenth century, or 'advanced sorcerers'- engrave our own desire upon our souls. I'm talking about magic names. We carve into our hearts the Latin for the reason we study magic, the one wish for which we would give our lives for. For me it's Fallere825, and Kanzaki's is Salvere000, nya~. The numbers afterwards are in case there're doubles of the same term. It's kind of like an email domain in that respect."
This is the same quote used by looper in his answer, but the official translation differs significantly. The official translation compares the inclusion of the numbers to an email domain rather than to email authentication. This means the two translations are offering completely different explanations for the numbers. In one case, the numbers are for the purpose of authentication so that others cannot repeat the name. In the other case, the numbers are simply for differentiating magical names that use the same Latin term.
It's easy to just offhandedly say the official translation is the correct one, but to really be sure about what's going on, we would have to check the original Japanese version. I will say, though, that the official translation makes more sense (though I do question the comparison to email domains rather than addresses). Kamijou hears the full magic names, so there is a flaw in using the numbers for authentication. If the purpose of a magic name is to tell opponents, then it doesn't make sense to give them the authentication as well. Also, at the point I've read to in this series, there hasn't been any mention as to why it would be bad that someone knew your magic name, so I question why authentication would be needed in the first place.