Webster is not complete about the meaning of the word arrondissement in French. This word is used for two different kinds of administrative divisions:

  1. A subdivision of the département (there are typically 2 to 4 arrondissements in one département).
  2. A subdivision of 3 major towns: Paris, Lyon and Marseilles.

For example, the département of Rhône contains two arrondissements of the first kind. One of these contains the town of Lyon, which itself contains 9 arrondissements of the second kind. No, it's not difficult to understand:

France -> 
  département of Rhône ->
    arrondissement of Lyon ->
      town of Lyon ->
        9 arrondissements inside Lyon

In Paris, the situation is even worse, since the département contains only one town: Paris itself. So the département of Paris contains only one arrondissement of the first kind, which contains only one town (Paris), which contains 20 arrondissements of the second kind.

In practice, arrondissements of the first kind, the ones Webster speaks about, are almost useless. French people, let alone tourists, rarely need to care about them. Arrondissements of the second kind are more important. For example, if you want to come to Paris as a tourist, it may be useful to know that a hotel in one of the first 6 districts will probably be very well located (and expensive, too).

Thanks jasonm for forcing me to be clearer.