I had never heard of it, but I found this description of arithmancy: http://dictionary.reference.com/browse/arithmancy So, an arithmancer must be a practitioner (one who practices) the activity described in the link.

This isn't just a fictional thing though. People did used to believe that you could assign numerical values to letters and words, and interpret the values in order to gain some magical insight about the words. Arithmancy basically refers to that practise, and a practitioner would be an Arithmancer. It is also used in the Harry Potter books, possibly slightly differently. It would be a mistake to say that -mancer means wizard, by the way. It comes from the Greek manteia, which means divination (i.e. gaining information about events through supernatural means), cf necromancy

Is this from Harry Potter? An arithmancer is someone who assigns magical properties to numbers and uses them for divination.

It's a word constructed from word roots. 'Arith-' means maths and '-mancer' means 'wizard'. I've never seen this word before. If it was in a fantasy novel it could be a literal wizard who casts spells based on numbers. Otherwise it could refer to someone who's so good at maths, it seems like magic. It's hard to tell exactly without context.