What is an odd function? I'm reading this term (odd function) in my numerical analysis book, but I have never heard of this. What does it mean that an function is odd ?

A function is said to be odd if changing the sign of the variable changes the sign of the function (keeping the absolute value the same). It is even if changing the sign of the variable does not change the function. We express this mathematically as:
Odd: $f(-x)=-f(x)$
Even: $f(-x)=f(x)$
It is sometimes useful to know that any function can be expressed as the sum of an even function and an odd function $f(x)=\left(\frac{f(x)+f(-x)}{2}\right)+\left(\frac{f(x)-f(-x)}{2}\right)$
The "odd"ness of a function is therefore an antisymmetric property, while evenness is a symmetric property. There are occasions in other contexts where it is useful to split something into symmetric and antisymmetric parts. So although the terminology may seem a little odd, it actually expresses something which is quite deep, and introduces an idea which can be fruitful. For example determinants are antisymmetric functions.
If the symmetric integral of an odd function exists, it is equal to zero, which sometimes saves a lot of work if you spot it - i.e. $\int_{-a}^{a}f(x)dx=0$ if $f(x)$ is an odd function. 
It is useful to know that the function $\sin x$ is odd, and $\cos x$ is even. The hyperbolic functions $\sinh x$ and $\cosh x$ are the decomposition of the exponential function $e^x$ into its odd and even parts.
