What does the series $\sum_{k=-\infty}^\infty a_k$ mean as appose to $\sum_{k=0}^\infty a_k$
Where $a_n$ is a complex sequence.
And how can one take the product of two such series (in a way similar to the cauchy product), for example the product of two complex valued fourier series.
The series $\displaystyle\sum_{k=-\infty}^{\infty}a_k$ is simply a shorthand for the sum of the two series $\displaystyle\sum_{k=0}^{\infty}a_k$ and $\displaystyle\sum_{k=-\infty}^{-1}a_{k}=\displaystyle\sum_{k=1}^{\infty}a_{-k}$. For Laurent series, these are respectively called the 'regular' and 'principal' parts. This should explain how your second series differs: it is missing the negative indices. In essence, we are summing over all integers but, of course, we can't start a summation at $\infty$ so we start at some finite $k$. The $\infty$'s are shorthands for limits, not real indices.
For more information, see Encyclopedia of Math: Laurent Series.
Regarding your second question about taking the Cauchy product of two series from $k=-\infty$, it's not very dissimilar from the form using power series; the only difference is in the indices.
For more information, see Math Stackexchange: Laurent series of...