There are many possible places to start in Special Relativity. That is, it is a matter of choice what is to be called a 'postulate' and what a 'theorem.' However, it is best to start with the most believable and simple. With this in mind, our first postulate states:
All inertial reference frames are equivalent.
In inertial reference frame (see the introduction to this topic for an explanation of reference frames) is simply in which Newton's First Law holds (that is, the law of inertia: all bodies remain at rest or in constant motion unless a force acts upon them). This means any non-accelerating reference frame is an inertial one. In an accelerating (including rotating) frame we have to invent 'imaginary forces,' such as the (coriolis or centrifugal forces) in order for the law of inertia to be valid. Our postulate then says that any reference frame at rest or in constant motion is as good as any other--there is no absolute frame. More precisely it says that an observer on the ground (or someone at rest in outer space) does not have a preferred or 'more realistic' view of events than an observer whizzing past on a train; their different ways of viewing and measuring things are equally valid. It makes no sense to say something is moving. It only makes sense to say something is moving relative to something else. For this reason this postulate is sometimes called the relativity principle.
The relativity principle also asserts that if the laws of physics hold in one inertial frame, they must also hold in all the others. This arises because there is no preferred frame: what is true in one frame must be true in all the others, otherwise one could pick a particular frame as 'special.' It is also crucial to understand that this postulate implies that if there are two frame F and F', then F must view events in F' in exactly the same was as F' views events in F. This has profound consequences for the laws of physics.
The second postulate states:
The speed of light has the same value in any inertial reference frame.
The first postulate was simple enough, but this one should be very worrying indeed. If a spaceship is hurtling towards you at 1.5×108 m/s and it fires a golf-ball from its nose at a speed relative to the spaceship of 1×108 m/s, then you would measure the speed of the golf-ball as (1.5 + 1)×108 = 2.5×108 m/s. However, if instead of a golf-ball the spaceship fires a photon of light at c = 3.0×108 relative to the spaceship, our postulate tells us that we would not measure the speed of the photon as 1.5×108 +3×108 = 4.5×108 m/s, but at 3.0×108 m/s, the same speed as in the spaceship's frame.
This may appear somewhat confusing. Clearly light does not behave like a golf-ball. No matter how fast the spaceship is traveling, either towards us, or away from us, or at any angle (so long as it remains in an inertial frame), we will measure the speed of the light reaching us to be c. Moreover, any nearby observer in any inertial frame will see the light traveling at the same speed. The veracity of this remarkable claim has been confirmed by numerous experiments, the first of which was performed by Michelson and Morley (referred to in the introduction to this topic); their careful experiments found no difference in the speed of light as the earth rotated around the sun (changing its direction of motion).