What type of data can we get from fMRI to calculate correlations? Lets say we want to compare the correlation between the activation one part of the brain and rise in value of other variable. The design would be within subjects.

What is the data that we can receive from the fMRI from one specific brain area(e.g. temporal cortex) to use to calculate the correlation.
The BOLD response is used in fMRI to provide a quantified measure of activation: https://en.wikipedia.org/wiki/Haemodynamic_response. It is based on the ratio of oxygenated vs. non-oxygenated blood, implying blood flow, and inferred to mean more processing or activation of those brain areas.