A 'system' is a process to put it simply. It could be making toast or computing an integral. It doesn't have to be complex. It is simply a list of instructions for any process.
A system is deterministic if the input determines the output. Or, if the present state predicts the future state. So, for instance, toast making is deterministic since we know that having it toasted for a certain amount of time results in different states. For instance, 5 minutes results in burnt toast!

Chaos theory is called deterministic chaos in technical terms. It is the theory about random behaviour in deterministic systems. This makes no sense at first (since an unpredictable system is known as stochastic) but it makes some form of sense in deterministic systems.

Let's suppose that the system is a square of a number. Then inputting 4 will obviously result in 16 since we know the present state of 4. This is a deteriministic system since we can predict the output easily. However, suppose that the computer instead worked out the cube of a number as a random error. We then end up with 64; very different from the predicted 16.

One "famous" example is Lorenz. He was running convection current calculations (back when a snail ran faster than a computer) and he decided to take a break from a long calculation so he printed the numbers out. When he restarted, the answers were totally different. It turned out that it stored more decimals than it printed. For example, 3.14199 was printed as 3.141 and when put back in would be 3.14100 which is totally different. This subtle difference led to the butterfly effect.

Since we know convection currents can affect weather (i.e. move clouds about), let's assume that we can use a deterministic model of a convection current. A flap of a butterfly can change an input (number) by a thousandth. This random error leads to an unpredictable outcome even though the system is deterministic. In other words, the future state of the weather would be completely different. This is why it is virtually impossible to completely predict the weather. We can accurately predict it for two days however.

Chaos theory doesn't mean that the weather is completely unpredictable. It just means that there is a deterministic system which is completely predictable for everything. In other words, imagine there is a formula for everything. However, in reality we can't access this formula. Thus, we have chaos theory.