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! :)

There are multiple ways to indicate a summation (similarly with a product, direct summation, etc.)  In all cases, the sum is evaluated in all ways such that the condition is met, subject to natural restrictions that are implicit in the problem.  Typical restrictions are that the index is an integer, and sometimes that the index is nonnegative.
Examples: $$\sum_{i\ge 0} f(i)$$
and $$\sum_{i\in[0,6]}f(i)$$
and $$\sum_{j|n} f(j)$$
and $$\sum_{j|n, j\neq 3} f(j)$$
and $$\sum_{jk=n} f(j)f(k)$$
and $$\sum_{j+k=n} f(j)$$
