Anyway, there are very mathematical ways to explain how CA's work, but here's Wikipedia's one. One way to look at CA is also as a kind of state machine with many different states at very short intervals from one another, where these states are actually macro-states, the global sum of internal states of each cell. Because rather small changes in internal states can significantly affect the global outcome of the global state, the horizon over which one can make calculations to derive future states is rather limited. I.e., one needs to calculate every state inbetween in order to find the final answer.
Some three centuries ago we started discovering/inventing physics laws and formulas to make our lives easier. Nowadays these laws and formulas were used to construct airplanes and we went to the moon with them. Most of these laws come with rather large assumptions. Most of the time, it is:"Assuming nothing happens that introduces a significant error, we can derive our future position/velocity/acceleration by multiplying x with y over a z time period". We're just lucky that macro-objects like our vehicles behave that way in a consistent manner.
But looking at smaller interactions or larger systems like the weather, we can't use those laws as directly as that. The number of collisions and forces between objects make the entire thing so complex, that you can no longer work with laws that require these assumptions. So the complication is that you now have to represent many other bodies interacting with your system and calculate the state of this "universe" or "world" for each intermediate state, until you get to the goal state you want. Luckily the interactions are not usually really complex when you get to an appropriate level. Unfortunately, exactly knowing these interactions remains difficult in many cases and very slight differences in the "rule" can eventually produce very large deviations from the overall pattern.
It is the expectation that this kind of thinking will produce more understanding about the world around us, as there are so many processes that function according to these principles:
- the billowing of smoke and vapour
- pressure of gas
- the way vortexes are produced by wings
- interactions between neurons?
- the structure of snowflakes
- the ways how cells react to other agents?