Presta vs. Schrader
Before you can buy a new pump you need to know whether you need a Presta or Schrader, which you can easily determine by looking at your tire valve. If it's a thick black valve like the one on a car, you need a Schrader pump. If it's a long, skinny metal valve with a nut on the top, you need a Presta pump. Pumps designed for Schrader will not pump up a Presta valve and vice versa.
If you have several bikes that use different valve types, or aren't sure what type of valve that you have, get a pump that is compatible with both Schrader and Presta. These pumps feature either separate nozzles for each valve or a simple control for switching from one to the other.
The floor pump is the most popular style of pump. Stand on the base and pump the tire by pushing the handle up and down. Many floor bike pumps include a pressure gauge so you can get an accurate pressure reading. They offer a fast, steady fill.
An alternative to the floor pump is the foot pump, which uses a spring-loaded mechanism that you pump with your foot. They're a less-expensive alternative to the floor pump and provide a good, steady pump rate. They often include a gauge. In general, foot pumps are less efficient and durable than floor pumps. Foot pumps usually use a Schrader valve.
Since it'd be impossible or at least very cumbersome to carry a floor pump or foot pump on your bicycle, mount a frame pump on your bike frame. When it comes to portable pumps, frame pumps are the largest option, so while they will provide the most efficient fill (still slower than a floor or foot pump), they are heavy for portables and don't fit easily on some frames.
Mini-pumps are smaller versions of frame pumps. They're the most compact, lightest pump option, and some even fit in a bike pack. However, they draw and pump a very small amount of air, and it takes a tiresome amount of pumping to fill a tire. In short, they're not fun to use.
Dual-action pumps, often found in mini-pumps, have a modified design from the average pump. Most pumps pull air into the chamber on the upstroke and then force it out into the tire on the downstroke. Dual action pumps are designed to pump faster for their size by forcing air into the tire on both the upstroke and downstroke.
CO2 pumps are an option if you don't want to spend the time and effort using a manual frame or mini-pump. They use a standard CO2 cartridge to fill your tire, providing a very quick, easy fill. Some CO2 pumps include on/off regulators so you can better control the fill, while others do not and will dump the entire cartridge into the tire. While CO2 pumps are even smaller than mini-pumps (though they do require cartridges as well) and are easy to carry, the main problem with them is that they're exhaustible. Once you run out of cartridges, you can no longer pump your tire, so pack enough cartridges for your bike ride.