Knowing how data was collected can tip you off to potential biases in the set.
All 15 executive departments of the U.S. federal government collect and publish data. Those departments are:
Other federal agencies and commissions may publish data of their own as well. A few sites to look at:
Typically this is to inform viewers of the subject of their operations.
What nonprofits will have useful data for will depend on your field of study. For instance, a medical student would want to search for datasets published by:
Corporations and business analysts will produce their own datasets to evaluate a company's performance or analyze trends within an industry. Some of these reports can be found through Business Source Ultimate (see below), but in general they are quite expensive and it is difficult for the library to acquire them, even through Interlibrary Loan.
Scholars in many fields produce data during their research. Some authors of scholarly articles will publish these datasets alongside the article, and others are willing to share the datasets if contacted by email. Some individuals will also upload datasets to repositories such as those listed below: