Why does time go forward? What's with the Second Law of Thermodynamics, and entropy, and all that? Why do I remember the past, and not the future? These are all really the same question.
We can think of the "direction" of time as being the presentation of the asymmetric nature of our memory; we remember the past, but not the future. Our idea of time is based on the process of cognition, and the observation that we can remember things on one side of the present moment, but only predict what is on the other side.
But why do I remember last week and not next week? Why do I experience a ceramic plate smash into a dozen pieces when it hits the floor, and never see it go the other way? The reason is the Second Law of Thermodynamics. This law says that the amount of entropy in the universe will always increase with time (entropy is essentially disorder...physicists and chemists and so forth can calculate and quantify the amount of entropy in a system, but it's pretty complicated).
So, the plate falls and smashes because it is going from an ordered state to a disordered state, and as far as the universe is concerned, that's a good thing. But, why does the Second Law of Thermodynamics make entropy increase with time, instead of decrease? Seems kind of arbitrary, doesn't it? The answer lies in the way our memory itself works.
A system's entropy can decrease as a result of a reaction, but only if that reaction makes the amount of entropy in the surrounding universe increase to a greater degree. For example, when you make ice cubes in your freezer, the freezer is removing heat from the water to such a degree that the water becomes solid. The water's entropy is decreasing (the solid state is much more ordered than the liquid state) but the heat which is released creates more entropy in the surroundings by making molecules in the surroundings bounce around more. There is a net increase in entropy in the universe, so all is well.
Our memory is another process which works in this way. Nobody really knows the gritty details of how human memory works yet, so for simplicity's sake I'll use the memory of a hard drive to explain. When your computer saves a file to its hard disk, the ones and zeros which comprise the binary code of the file are converted to different arrangements of magnetic domains on the surface of the disk. This makes things more ordered on the disk, which decreases the total entropy of the disk. However, it takes quite a bit of energy to do all this sorting and disk-spinning and saving and so forth. A lot of this energy is lost as heat, the heat makes the molecules in the surroundings jump around faster, and in the end, the entropy in the universe is increased. And in fact it couldn't happen any other way, due to the nature of the process.
Human memory is a bit different, but the same basic principle applies: energy is expended to bring order to a system, and the increase in entropy due to heat outweighs the decrease in entropy due to a more ordered group of neurons. Furthermore, the principle applies not only to short-term and long-term memory, but also to any process of pattern recognition or pattern forming in the brain. In short, it applies to the whole process of cognition.
So, memory and cognition are based on the Second Law of Thermodynamics. We only remember the past because the process of storing a memory is one which increases the amount of entropy in the universe. Any future event would occur in a universe with more entropy than the present universe, thus negating the possibility of any process between now and then (or ever) which would let me "remember" it.
Furthermore, the Second Law of Thermodynamics is based on the direction of time; as time moves "forward", entropy increases.
And, of course, the direction of time is based on our memory and cognition; the past is what we remember, the future is what we don't remember.
From this we can see that these three concepts of time, entropy, and cognition emerge co-dependently and are really just different aspects of the same thing.
This write-up was informed and inspired by a chapter in Stephen Hawking's A Brief History of Time called "The Arrow of Time". Great chapter, great book. Read it.