Rotary and Hover Mowers
Both of these whack the grass with a horizontal spinning blade, efficient for the average family garden as well as rougher kinds of lawns. Rotary mowers have wheels; hover mowers are mini-hovercrafts.
Rotary mowers
All rotary mowers have a blade turning horizontally on a vertical spindle within a protective canopy. The difference between a rotary mower and a hover mower is that the rotary mower has wheels. Generally, rotary mowers are rather more robust as well - and in some cases very robust indeed. They do not give such a good cut as cylinder mowers, and tend to tear the grass, producing a slight browning effect on the tip of each blade of grass, especially if the mower's cutting blade is not super-sharp. Rotary mowers are, however, good for lawns that have to stand up to hard usage ('utility lawns'), such as the typical garden lawn, where children might want to play games.
Into the long grass
You can adjust the height of the blade (usually by means of a lever system connected to the wheels, on by adjusting a frame within the chassis) to match the length of grass that you desire. A real advantage of a rotary mower is that you can raise the chassis on the back wheel to lower it onto really long grass - not something you can do with a cylinder mower. In other words, you can tackle a meadow with a rotary mower.
Roller
Some rotary mowers are fitted with rollers, to give that striped effect - but it has to be a pretty heavy roller to be effective.
The box
Larger rotary mowers come fitted with a grass box: this is a mesh affair, which permits the clippings to be blasted out of the protective canopy on a jet of air. (For the debate over whether or not to use a grass box, see "Box or no box?" in the separate article on Cylinder Mowers.) However, the mesh of rotary mower boxes tends to get clogged up with debris, reducing their efficiency dramatically. So you should clean the mesh regularly, by removing the box and brushing it, or subjecting it to a jet of water from a hose, or (for encrusted debris) by soaking it in a bin of water and washing detergent, and then brushing.
Mulching
A rotary mower with a 'mulching function' has a specially shaped blade that chops the clippings very finely; the machine then scatters them on the lawn, rather than collecting them. This is a very effective way of returning nutrients to the grass, and helping to retain moisture in the lawn, provided that the lawn is cut often and the quantity of mulch is not excessive.
Petrol or electric?
Rotary mowers can be either petrol driven, or electric. Electric rotary mowers usually power only the rotating blade, not the wheels, so the operator has to push the mower forward. For the debate over which power-source to use, see the separate article on Petrol or Electric?
Hover mowers
Hover mowers are a by-product of the invention of the hovercraft in the 1950s - and indeed this invention seems to have survived better in its application to lawncare than as a means of transport. The cushion of air created within the plastic hood of the chassis keeps the rotating blade off the ground - so hover mowers don't need wheels. Almost all hover mowers are electric: they are lightweight and manoeuvrable, and are particularly good at dealing with grass where other mowers dare not venture, such as steep banks. Because hover mowers have no wheels, they can be swung from side to side, in the same manner as a strimmer - but they are a lot quieter. The height of the cut can be adjusted in some models by removing or adding spacing washes on the spindle above the blade.
Maintenance
Apart from keeping the mechanisms oiled, and the interior of the chassis free of debris, the key thing is to have a sharp blade. This applies to both rotary moweres and hover mowers. You may be able to sharpen your blade (or rather the two cutting edges of the rotating blade) with a file, angle grinder or carborundum stone. If the blade has become very blunt, or chewed up from hitting rocks (it happens), you should consider investing in a new one. This should not be expensive, and generally the blades of rotary and hover mowers are easy to replace. Note that a really chewed up blade is likely to be unbalanced, and can therefore do serious damage to the motor.