With all the attention that Vespa, Honda mopeds, and Yamaha scooters get, some other quality brands are often overlooked. You can find many of these used mopeds for great deals.  So, they are cheap mopeds, in the sense that they are affordable, but they are reliable and solid.  Here is a short list of other great scooter manufacturers, and the some models they are popular for:

Wards, Model 57008
Wildfire WF-20MB, WF-24MB, WF-26MB
AMF Roadmaster
Aspes
Balboa
Baretta Model 49cc
Batavus
Benelli Blazar
Bermuda 147FA
Califfo
Carabella Motomatic 49cc
Cimatti City-Bike
Clinton 50-A
Columbia Commuter, Imperial, Medallion 2271, 2281, 57062, 2251, 2241, Motrek, Western Flyer
Com Ute
Concord XKE, Blazar, Colt I
Derbi C-5 Diablo, Laguna Sport, RD-50, S/L, T/T
Eagle 1, 3
F. Morini
Flandria 101, 102
Gadabout
Garelli Basic, Rally SL, Rally Sport, Super Sport
Hercules P-1, M-4
Indian AM 1-50
Intramotor Blanco, Pinto, Scout

	2009 SYM MIO 50 Urban Scooter - 50cc 4-Stroke	$2,295.00
	2010 Peugeot Satelis 250 Premium 4 Stroke Single Cyl.	$6,995.00
	2009 Peugeot Speedfight2 Ultimate Edition 50cc 2-Stroke	$3,750.00
	2009 SYM RV 250 Pleasure Tripper - 250cc 4-Stoke	$4,999.00
	Lambretta 1955 - Model D 150cc (Completely Restored)	$2,025.00
	Blata Elite 11 - Mini-Bike - Great Condition	$499.00 $549.00
	2004 ETON Beamer 50cc Scooter.	$550.00
	NEW YAMATI CHROME EDITION 150CC SILVER SCOOTER RX150XE	$999.99 $1,100.00
	NEW YAMATI CHROME EDITION 150CC BLUE SCOOTER RX150XE	$999.99 $1,100.00
	1976 Garelli moped vintage 50cc italian	$299.00

Technically, used mopeds are defined by limits on engine displacement, speed, and power. Usually cheap mopeds are restricted to 18-47 miles per hour in speed. The max engine size is usually 50cc. Some places require pedals, thus making them a form of hybrid bicycle/power vehicle,. In some countries, the legal driving age for a scooter is lower than for big motorcycles, which is why you'll see a lot of young people driving them around.