If f is continuous on a closed interval [a, b] and f(a) and f(b) have opposite signs, then there exists a number c in the open interval (a, b) such that f(c) = 0.
In short, if a function is continuous between two points, and one is above the x axis, and the other below, there is a point which crosses the axis. Simple, isn't it?
Amongst other things, Bolzano Theorem lets us, for example, have a (simple) numeric iterative method for narrowing zeroes on a function. Say you've got point a which is positive, and b which is negative. There is a zero between them, so let's find the image of (a+b)/2. If it's positive, the solution is between (a+b)/2 and b, else between a and (a+b)/2. Thus you can go on halving the interval (though I'd suggest using Newton-Raphson method instead).
Now, an interesting exercise. Suppose that the temperature along Earth's equator is continuous. Prove that at any given time, there exist two opposite points on it which have the same temperature.
Solution:
Let's define f(x), which is the temperature on Earth's equator. x is the relative position on the equator. Say that between x=0 and x=1 we have the whole circumference. Note that this function is periodic (i.e.: f(x)=f(x+1)).
We want to find an x which makes f(x)=f(x+1/2). This is the same as finding zeroes of the function g(x)=f(x)-f(x+1/2).
Now, we calculate g(x) for the following x:
x=0: g(0)=f(0)-f(1/2)
x=1/2: g(1/2)=f(1/2)-f(1)
Note that f(1)=f(0), according to the above:
g(0)=f(0)-f(1/2)
g(1/2)=f(1/2)-f(0)
g(0)=-g(1/2)
Now, g(0) and g(1/2) have opposite signs. Therefore, by the Bolzano theorem this function has a zero between 0 and 1/2, and then we have found a solution for f(x)=f(x+1/2).