There's more than a little confusing regarding paintball guns and what types there are in the world. This is intended to help people understand differences between types of paintball guns.
Pump Paintball Gun
A pump paintball gun is the most basic kind of gun available. It is a very basic gun wherein you have to pull a pump handle forward and backward after each shot to seat the next paintball and prepare the gun to be fired. It is the original paintball gun design and is a very simple, reliable gun. Pump paintball guns are not nearly as common now as they were a decade ago, but they some players still do use them, particularly in stock-class paintball events.
3-Shot Burst3-shot burst (also known as 3-round burst) is a firing mode where one pull of the trigger will result in three shots being fired. This type of firing is typically found on electropneumatic paintball guns that have multiple different firing modes (meaning that you can switch between 3-shot burst and semi-automatic). 3-shot burst is not particularly useful in paintball as most players will stick with either semi-automatic or assisted firing (ramping or full-automatic).
RampingRamping is a firing mode that requires the trigger to be pulled consistently but a circuit board will gradually increase the rate of fire. For example, pretend that ramping is set to kick in at 4 pulls per second. This means that if you pull the trigger at a rate of three times per second, the gun will just continue to fire at a rate of three times per second. If, though, you start to pull the trigger at a rate of four balls per second (or faster), the gun will initially fire at four rounds a second but will gradually increase the firing rate (it "ramps" up the firing rate) as long as you pull the trigger. This means that a player may pull the trigger four times a second but the gun will gradually shoot faster and faster until it reaches its maximum rate of fire (which may be 20+ balls per second). This firing mode is legal in some tournaments but not in others, so be careful before you take it to an event.
Fully-AutomaticFully-automatic paintball guns require you to pull the trigger one time and as long as you keep the trigger depressed, the gun will continue to fire. Fully-automatic guns have a defined rate of fire that varies by gun. Most tournament and many fields prohibit fully-automatic paintball guns.
Machine Gun Paintball Guns"Machine Gun" paintball guns don't really exist. This is a commonly used term that usually comes from someone who is unfamiliar with the sport. Generally, when someone refers to a "machine gun" they are referencing a paintball gun that shoots very quickly such as a gun that has a fully-automatic or ramping mode.
There are paintball guns that are designed to look like actual machine guns. Many of these, though, are just semi-automatic guns.