What is the difference between the FYEAR and the DATADATE. When we extract the year from the datadate, it is not always the same as the fyear. Why is this? When examining across time periods, which should we use?

Usually the student should make a download and determine from the data what is going. FYEAR indicates the fiscal year that the company is in at the time of DATADATE. When using the annual Compustat it is pretty simple to understand, DATADATE gives us the annual close of fiscal period, in the case of MSFT (June FY) this might be 6/30/2010, and the FYEAR is 2010. In a more granular download (such as quarterly) we find DATADATE/FYEAR as:
DATADATE FYEAR FQTR
20100331 2010 3
20100630 2010 4
20100930 2011 1
20101231 2011 2
20110331 2011 3
20110630 2011 4
You could then want to use PRCC_F to get the closing price for the fiscal year.