Everything inside the "Boulevard Périphérique" (as answered by DJClayworth) can be considered city center.
But, a lot of people (tourists) consider the 7 most centered arrondissements (borough) as the city center because they contain most of the tourist attractions (museums...)
Every accommodations in those areas will be more expensive.
As you get further from those arrondissements, you will find cheaper accommodations (rentals, hotels and hostels).
If you stay inside the "Boulevard Périphérique", then you will have no difficulty moving around, you will be at walking distance to a subway station or bus that will drop you wherever you want.
If you decide to get outside to the Banlieue, something like Vincennes or Nogent-sur-Marne (the only places I really know outside the center); then be careful that you have access to proper transport (bus, metro, train).
When doing that, you need to plan your transport cost and all the "wasted" time spent in subway/train.
My sister lives in Nogent-sur-Marne, she has access to 2 trains that goes to Paris (RER A and RER E) and that takes about 30 to 45 minutes on a good day to get from there to the Louvre Muséum.