There are two words that can be translated into English as 'moped'. The first is a bromfiets (or brommer). They have a registration plate with black writing and yellow background. They have a maximum speed of 45 km/h, and cylinder capacity of not more than 50cc, or electric motor with maximum net power of not more than 4kW. The second is a snorfiets, which is legally a subtype of bromfiets. Snorfietsen have a registration plate with white writing and blue background. They have a maximum speed of 25 km/h. A class AM driver licence is required for both. Both types of mopeds must be insured.
Scooter is not a legal category; it can be a motorcycle, a bromfiets proper, or a snorfiets.
The slower snorfietsen follow the rules of bicycles, unless otherwise indicated. Similar to bicycles, helmet is not compulsory for snorfietsen, and snorfietsen usually use bicycle paths. (However, the central government has made it possible for municipalities to make snorfietsen follow the rules of bromfietsen, i.e. wearing helmets and banned from bicycle paths. See the second figure in this cyclist blogpost for the change. So far there has been talks of such restrictions in Amsterdam, but it is currently (September 2016) not in place. Check what Amsterdam says from time to time.)
Snorfietsen ride on compulsory bicycle paths (see sign G11 in the cycling page). On non-compulsory bicycle paths (see sign G13 in the cycling page), snorfietsen with electric motors are allowed, but snorfietsen with combustion motor must have the combustion motor off in order to use non-compulsory bicycle paths.
The following are some bromfiets-related signs that you will see in the Netherlands. (In the figure captions below, 'moped' means bromfiets proper, unless otherwise indicated. 'Bicycle' includes both bicycles and snorfietsen). See also the pages on cycling, motorcycling, and driving for more on Dutch traffic rules.
Snorfietsen must not be capable of going faster than 25km/h, and 25km/h is the speed limit everywhere. Bromfietsen must not be capable of going faster than 45km/h, and 45 km/h is the speed limit, except on moped/bicycle-paths, where the speed limit is 40 km/h outside built-up areas, and 30 km/h inside built-up areas. (See the driving page for the distinction between outside and inside built-up areas.)
Utrecht, Amsterdam, and Leiden have plans to prohibit bromfietsen and scooters with bad emission levels from entering their milieuzone 'environmental zone' in the comming years. See what ANWB says on this.