What is the city center of Paris?

We don't do specific recommendation on this site, so the best we can do is give you some ideas of places to look for cheaper accomodations.

For purposes of expensive accomodation, you can consider the centre of Paris to be the 1st Arondissemont, essentially the centrepoint of this map. Anything in the area of the Louvre, the Arc de Triomphe and the Eiffel Tower would be 'centre ville' as they say.

Your accomodations are going to be cheaper the further you get from here. Being near a metro station makes it easier to get into the centre, but places near a metro will be priced higher than those that are not. You are not going to find anything like 'cheap accomodation' within the Boulevard Peripherique - the ringroad around the centre on the above map. 'Normal accomodation; can be found just within that - depending on your definition of 'cheap' and 'normal'. However Paris transit is pretty good, and getting into the centre for a day from outside the Peripherique is perfectly doable. It should also be said that Paris is a pretty expensive city - even your cheaper accomodation is going to be more expensive than other less-well-known places.

Paris does have its share of hostels, and if you book far enough in advance, and don't mind the lack of facilities, you may be able to get something within your price range.

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.

Anecdotal:

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.

The centre would be Place du Châtelet, which also boasts the main transport hub of the city, Châtelet (Metro) and Châtelet-Les Halles (RER).