How do paper planes work?

At a basic level, the flight of all aircraft is determined by four forces: lift (upwards force), thrust (forwards or 'accelerating' force), weight (downwards force, the force of gravity) and drag (backwards or slowing force). Paper aeroplanes are no exception, so this is a good place to start. They are un-powered after they are released, so they experience zero thrust during flight. As they are heavier than air, this makes them gliders, so their flight is best described as gliding (or perhaps falling with style).

Furthermore, the wings of paper aeroplanes provide very little lift. In large aircraft, the wings generate lift by exploiting Bernoulli's principle--a fluid (such as air) will have lower pressure the faster it moves, and vice versa. This is done primarily through the wing shape and the angle of the wing (the angle of attack) causing air to move over the wing faster than under it, causing a pressure differential which makes the wing rise. Do note, however, that the reason for the air moving over the wing faster than under is complicated and is not due to the air going over having to 'catch up' to the air going under (sometimes referred to as the equal transit-time fallacy), which is what I was taught as a kid!

Paper aeroplane wings are typically not curved (cambered) the way mechanical aeroplane wings are. While slight cambering can help provide more lift, not much cambering is required for the wings to generate a lot of drag, and the thinness of the paper minimising drag is a large reason for paper aeroplanes being able to stay airborne for as long as they do. Paper is also light, which helps to minimise the weight force that acts against the lift. 

While paper as a material clearly has some positives when it comes to flight, there are also drawbacks. For example, because paper is weak and most paper planes are constructed primarily by folding, the aspect ratio of paper aeroplane wings generally has to be very low. The aspect ratio of a wing is (very roughly) the ratio of its span (how far it stretches away from the body) to its chord (how far it stretches parallel to the body, see Fig. 1). Low aspect ratio wings, like those on paper aeroplanes, can also be found on fighter jets, while high aspect ratio wings are more common on commuter aircraft. This is because lower aspect ratio wings are generally better for faster flight and a higher aspect ratio is better for slower flight, so a slow paper aeroplane with low aspect ratio wings is disadvantaged. 

Figure 1: Examples of (a) a low aspect ratio wing and (b) a high aspect ratio wing.
The spans are denoted by $b$ while the chords at the fuselage are denoted by $c$.
In general the aspect ratio $AR$ is given by $AR=b^2/S$ where $S$ is the plan area.
Note that this reduces to $AR=b/c$ for a rectangular wing.

While the four forces of flight provide a good introduction to how flight works at a basic level, there are many other factors that come into play. One such factor is the stability of the aircraft, which is how well it handles small disruptions. At its simplest, an unstable aircraft will exaggerate the effect of a small disturbance, while a stable aircraft will naturally return to level flight. At neutral stability, a small disturbance would not be exaggerated but the plane would not rectify itself either; it would just continue onwards in the new direction. 

A good paper aeroplane must be stable or it will likely not be able to regularly glide very far. Many paper aeroplanes naturally have a lot of paper folded up near the front of the plane, which helps shift the centre of gravity (the balancing point) forward of the neutral point (the place where, if the centre of gravity were located there, the aircraft would have neutral stability). This helps give the aircraft stability, though too much weight forward will keep the nose down and paper aeroplanes don't have elevators to counteract this. This is only one type of stability (known as longitudinal static stability) and there are a number of others which good paper aeroplane design naturally accounts for. 

One such example is dihedral (bent-up) wings, which give the aeroplane lateral (roll) stability and prevent it from turning onto its back or spiralling, something which anhedral (bent-down) wings may encourage. Another example is winglets (small fins on the wing tips), which provide directional stability, i.e., they help keep the aeroplane headed in one direction, which can help compensate for the lack of a tail fin. For practical purposes, however, adding winglets can induce considerable drag (and thus actually reduce stability) if not angled straight-on. I won't go into more examples here, but as always I encourage the interested reader to pursue the topic further (there is a lot of very approachable material out there!). 

I hope this post has gone some way in explaining how a paper aeroplane flies. I think the best summary of what we've covered here is that it is very similar to how a full-scale aeroplane flies, but dramatically simplified. This is in retrospect somewhat obvious but, to me at least, it was very surprising! A final note: This is an excellent question, and when it was posed to me I couldn't give an answer that I would consider satisfactory (I still have some niggling doubts!). Since then I have gone to a number of Internet sources, listed below, from which I have drawn much of this post. I also solicited the help of my friend Kevin Yost, who knows a good deal more on the topic of aerospace engineering than I do. If you are interested in finding out more, I encourage you to do your own research, but be warned that most of what is out there is lacking in detail, so you may have to do some trawling.

Paper Airplane Aerodynamics

The Science of Paper Airplanes

Paper Aeroplanes and Wing-Tip Fins (this site may not be safe)

Wikipedia also has a number of very good pages on aerodynamics