You should be familiar with the exponential function ex. If you're not, look back at the MathHelp notebook on exponential functions.
Here the "e" just refers to a particular number, about 2.7 in fact. Because this number occurs very frequently and can't be written down exactly as a number (the decimal version goes on forever), it is usual to call it "e", by which we mean that number exactly.
For any value of the variable x, we can work out the value of "e to the power x", which gives us the value of the function ex. For example, if x=1 we get e1 which is just e, so it's about 2.7. If instead we consider x=2, we get e2, so that's about 2.7 squared, which is about 7.
Of course if x=0, we get e0, which is 1 (anything raised to the power zero is 1). So we're seeing the following behaviour from this exponential function ex:
As x increases, the function ex increases more and more rapidly (in fact its rate of increase at any value of x is equal to the value of the function at that point).
As x decreases, the function decreases, getting closer and closer to zero but never reaching zero.
So overall we get the following shape for the curve of the function:
If instead of ex we look at the function e-x, the picture is simply reflected about the y-axis (since we're just replacing x with -x everywhere). So we get this picture for the function e-x:
All this should be familiar to you, so you're probably asking what does it have to do with hyperbolic functions. The answer is in the next section.