What does $p+q=k$ mean in the index of summation? I need help solving something I don't understand. OK so the problem is this:
$$H^k(X,C)=\bigoplus_{p+q=k} H^{p,q}(X),$$
What does the $\;p+q=k\;$ mean? 
Thank you anybody that helps! :)

$p+q = k$ indicates that the sum of indexes $p, q$ is equal to $k$, where $k$ is constant, for the given $k$.  If $k=4$ you can take $p=1, q=3$  or $p = q = 2,$ etc. 
So every possible pair $(p, q)$ for which $p+q = k$; where $p, q$ are the indices which indicate which $H^{p, q}(X)$ to sum. 

From comment
Indeed: to "spell it out the notation" you've posted, for a particular value of $k$, let's let $k = 4.\;$ Then:
$$H^4(X,C)=\bigoplus_{p+q=k} H^{p,q}(X) = H^{0, 4}(X) \oplus H^{1, 3}(X) \oplus H^{2,2}(X) \oplus H^{3,1}(X)\oplus H^{4,0}(X)$$
