A horizontal asymptote is a fixed value that a function approaches as x becomes very large in either the positive or negative direction. That is, for a function f(x), the horizontal asymptote will be equal to lim_(x->+-infty)f(x).
As the size of x increases to very large values (i.e. approaches infty), functions behave in different ways. Some simply get bigger and bigger forever (e.g. x^2). Others oscillate up and down (e.g. sinx). But others get closer and closer to a particular constant value. For instance, consider the hyperbola y=1/x.
When we input very large values of x (think of x=1000, x=1000000, x=1000000000 etc.), the value of 1/x becomes very small. It gets very close to 0. And if we plot points to see this visually, we find that the graph of y=1/x approaches the line y=0. We can observe the same effect when putting in very large negative values of x, too. So we say that y=0 is the horizontal asymptote of y=1/x.
If we shifted the entire graph upward, say y=1/x+5, then we would find that the graph approaches the line y=5 instead. That's because lim_(x->+-infty)1/x=0 and so all that's "left over" when considering the horizontal asymptote is the 5.
A couple of important notes. Firstly, even though the graph approaches the asymptote, it will never get there. There is no value of x we can input into y=1/x that will make y actually equal to 0! So even though the graph approaches a certain value (like x=0), it's critical to understand that "asymptotic behaviour" means you never actually get there.
Secondly, horizontal asymptotes tell you how a graph behaves at its extremities (i.e. as x->+-infty). It doesn't tell you anything about how the graph behaves "in the middle", near the y-axis. For instance, going back to our example of y=1/x, the graph has nothing to do with y=0 for small values of x. In some instances, such as y=x/(x^2+1), the graph can even cross the horizontal asymptote. This is a critical difference between horizontal and vertical asymptotes! (Vertical asymptotes are a different thing entirely, even though they look very similar.)
