As a participant in modern traffic, you have to share the roads, paths and alleyways with hordes of other trafficants. If you are a "soft participant" in traffic - such as a pedestrian or a biker, and to a lesser extent a mopedist or a motorcyclist - there is a lot to think about.
The bicycle is unique in traffic
Busses, tractors, articulated lorries, cars, trucks, pedestrians, motorcyclists, road construction vehicles, playing children - they all make up a whole. Unique among all of these are the cyclists. Sure, pedestrians and playing children are undoubtedly more vulnerable in traffic, but there are a lot more of them, and their presence is frequently a lot more predictable. Bicycles, on the other hand, are special in that they often are the most silent of all traffic participants. They are also a lot faster than many of the other road users, but are simultaneously easy to overlook. Because of all this, riding a bike in traffic is a challenge: A dangerous one, if you don't know what you are doing.
Some countries - especially Holland, where there are more than 17 million bikes between its 16.5 million inhabitants - have excellent facilities for bikes and riders: Bicycle paths run everywhere, and the other traffic participants are used to be on the lookout for fast, silent twowheelers.
In other countries, however, such as urban UK and most of the US, people are not as accustomed to bikes being part of traffic, and might as such forget about them, or even ignore them completely.
A bike is completely different from a car
To anyone who has ridden a bike as a form of transportation, this is obvious, but sadly, many people only learned how to ride a bike when they were 5, rode to school a few times, and have since forgotten about it: A bike is the exact opposite of a car.
Whereas a car is most precise and stable when driving slowly (that is why you slow down when you come to a narrow gap, or when you have to park), bicycles work the opposite way: When riding very slowly on a bike, it becomes unstable. There are many reasons for this, but the short explanation is that the centrifugal force of the wheels works as a gyroscope, which means that the wheels tend to stay stable. This again means that a bike at speed is a lot more stable than a bike riding slowly. If you have ever tried to learn how to ride a bike with your hands off the steering wheel, you'll know how it works.
Because of this, cars are often likely to go around bicyclists at speed with a large arc. Fantastic, but not really necessary: As long as the speed difference between the overtaking car and the bicyclist isn't too ridiculous (or else things like drag starts having an effect), a car can easily stay in its own lane to overtake a bicyclist keeping to the edge of a lane. Conversely, a car who sees a bicycle moving very slowly, is likely to careen past the bike with a minimum of clearance: Obviously a dangerous situation, as the bicyclist is a lot less likely to be stable.
As a bicyclist it is often difficult to do anything about this: Many drivers are completely ignorant of the challenges of riding a bike, and in addition, many car drivers will be downright annoyed at "those pesky bicyclists", who often go significantly slower than the speed limits.
Defensive biking
The only way around it, then, is to participate in some defensive biking. In many countries (the Scandinavian countries, for example), it is encourage to bike on the pavement. This demotes the bicyclists from being a "proper" traffic participant, but it does help in changing the bicylist from a prey (to the cars) to a predator (to the pedestrians). Depending on the traffic situation, and depending on how predictable the pedestrian traffic is (drunk people and children are unpredictable, whereas commuters walking from the train to their places of employment are likely to stay on "their side" of the walking paths), it may be better to use the edge of the road in any case.
In other countries, riding a bike on the pavement is illegal - but the same applies: If there are a lot of nut-job drivers out there, and the pavement is empty, nobody will book you for breaking the law in the name of road traffic safety.
When dealing with cars on the road, proper signaling is important (point in the direction you are going, or show an up-down arm movement when you are slowing down or stopping), but it is equally vital to keep in mind that many drivers won't know what you are intending to do, because they do may know what the signals mean.
Additionally, it is quite important to keep a look-out on pedestrians. Many pedestrians - especially in places where bicycles are less common - will assume that because they can't hear a car or motorcycle, that the road is clear. As such, they could suddenly start crossing the road, and become the victim of an accident. By regulating your speed and / or calculating in a little bit of extra safety distance, it will give you the time to brake, to yell / use your bell, or take evasive action. Or all of the above, in a comical display of wailing, tyre squeal and wild jerking of the steering wheel.
Overall, if you are a bicyclist, the best thing to do is to assume that the cars and other vehicles around you are out to kill you, and that all other bicyclists and pedestrians are suicidal.
Oh, and always wear a helmet. Statistically, you are very likely to be in an accident, and on many bi-wheeled vehicles (bicycles included), your head is liable to be the first point of impact with whatever you ended up crashing into.
Most of all: Be careful.
Source: Common sense, and the dislike of frequently nearly getting killed on my daily 5-mile commute between Liverpool city centre and Bootle new strand.