Whether or not you are permitted to use general websites is up to your individual teacher, so before you do your research check with them. Then, you want to find the best resources possible for the assignment.
Although Databases will always guarantee you quality/academic resources, websites should not be overlooked, particularly when it comes to pop culture type topics, which are difficult to find on academic database sites. Many websites (particularly .org. .gov, and .edu) have valuable information to offer. For websites, however, keep in mind that you will need to do some checking for validity by researching the website's author, date of publication, biases, etc. Check out the "Verifying Source Validity" tab on this LibGuide if you'd like help with that.
(some of the tips have been included below)
If you need specific types of resources, like maps or polls, etc., you can use the .ORG search mentioned above to help you search out those specific resources. See the following screenshot (which shows how this individual wants information about poverty, but is specifically looking for polls and maps that contain that information).
If you'd like to explore a LibGuide dedicated to research papers, then check out the Research Papers & General Research LibGuide (complete with Databases, Websites, Debate Sites, and Additional Tips).