Could someone please explain tome what an arrondisement is. We will be traveling in October and am trying to plan.

An arrondisement is a neighborhood, so to speak. Each one has their own flavor. If you think of Paris as the USA, each arrondisement would be a state. Break it down further, state:county, county:town, etc etc.

I kind of look at it the way New York City is broken up into burroughs, to be honest. It's not really as regimented as a state in the US.

It is a political subdivision of the City. I believe the reason they are called "arrondisements" has to do with the shape in which they were drawn--in a circular manner spiraling out from the center of the city. They also appear in French zip codes--e.g., the postal code for the 4th arr. is 75004, for the 5th it is 75005, etc.

An arrondissement is an administrative district, usually somewhat larger than what you might call in your hometown a "neighbourhood". The arrondissements in Paris are organised in a snail shape like this: http://ssa.paris.online.fr/

The 1st, 2nd, 3rd and 4th arr are fairly small ones on the Right Bank (north side of the river), closest to the river. 5th, 6th and 7th are on the Left Bank (south side of the river), closest to the river. Then you're back to the Right Bank again with 8th, 9th, 10th, 11th and 12th. The 13th, 14th and 15th are on the Left Bank, and 16-20 are on the Right Bank.

Each arrondissement has a different reputation and character. The 1st-8th are probably the most popular with tourists because these areas are central, close to popular sites, etc. Some of the arrondissements have a reputation for being a little more downtrodden or dodgier than others (e.g. 10th-14th, parts of the 15th, and 18th-20th), but some areas can be great for bargain hunters. The 16th and most of the 17th, however, are rather upscale. Parts of the 16th (those closest to the Eiffel Tower) are quite expensive, but may have better views than those on the Left Bank right under the tower.

A lot of cities are divided into different areas that have numbers or some other way of distinguishing them. It's very common -- you have the same thing in London, Prague, Vienna, etc. etc. It's just a part of the city. They are called "districts" in some cities. (I don't think it is analogous to US states because arrondisements aren't broken down into further political subedivisions like counties, towns, etc., as states are). It's just a way of enumerating different parts of the city, or referring to them.