It is just that this question gets answered a lot and, really, it needs an understanding of special relativity to understand the answer. It is much easier to go on to Wikipedia than for me to try to cobble together an explanation. But, here goes...

The theory of relativity is consistent with measured reality and, in particular, that the speed of light is the same for all observers. This seems contrary to observations of moving objects in everyday life, but is nonetheless true. The effects of this are not really noticeable in our normal life because we don't ever encounter such high speeds (light speed in a vacuum = 186,000 miles/second) however there are many cases nowadays where the theory of relativity has to be used because of the precision required. And, indeed, speeds close to light speed are achieved by particles in particle accelerators and by cosmic rays coming from space. In both these cases relativity theory has to be used.

The laws of motion are changed as a consequence of relativity although, at everyday speeds they approximate closely to those of Isaac Newton. A seemingly bizarre consequence is that there is no absolute measurement of the passage of time - if you observe someone who is travelling relative to you then, from your perspective, his clocks would be running slowly compared with yours. Similarly, to him, yours would be running slow. This may seem paradoxical and it is not easy to understand (see the Twin Paradox) but there is, in fact, no paradox. Another consequence is that distances, in your direction of motion, contract (see Lorentz contraction) or, from the perspective of someone watching you pass in your rocket, he would measure your rocket as shorter than you would.

I'm afraid none of these things seem reasonable based on everyday experience but they are true and have been verified by numerous experiments. I'm not good at writing equations on this website but Wikipedia has them if you wish to see the maths behind it.

To go to the half light speed ship; if you fire your light out the front a stationary observer would see the light travel at exactly light speed (no faster). He would see the light change colour to be more blue because of the Doppler effect but the light speed would not be different. Perhaps more interesting is if you fired a particle out also at half light speed with respect to yourself. A stationary observer would see the speed of this particle as 0.8 x lightspeed (if I've done the maths correctly). No mass can be accelerated to the speed of light however you try it - another consequence is that the energy required to accelerate a mass to the speed of light is infinite. Light (photons) can only exist at light speed and are sometimes referred to as having zero "rest mass".

For the non-physicist who wants to understand these things, I think a key issue to grasp is why, in the fist place, we expect things to behave as we commonly observe them when taken to extremes. We are only really built to have intuitive understanding of the world around us sufficient to help us survive and reproduce. There is really no reason why the world should always fit in with the limits of our ability to intuively grasp how it works. Special Relativity is a good example. It leads to the consequence that space and time should be considered as a 4 dimensional whole and that time is not absolute. Distance and duration are not independent but the universe should really be measured as "intervals" between "events" in Space-Time. General Relativity shows that Space-Time need not be "flat" and is curved by the presence of matter. The mathematics of what happens in this case gets very difficult to calculate. And Quantum Mechanics tries to explain what happens and the fundamental level within the matter that makes up the world. Nobody has yet united the two extremes in one consistent theory.

I hope this is more helpful. It took me an hour to write.