You'll run into some professors who won't let you use websites, or any materials you can get from the web even if they are scholarly, and you'll have some professors who let you use anything. Even if you may be able to use websites for your work, this doesn't mean that you don't have to critique the sites that you use. Because you do. In fact, you should be critical of everything you use while in college.
Below is a very simple list of criteria that you should use when judging a website. The link goes to a really good site that goes into more depth on this process.
Some of your assignments and research papers will require the use of a website - but the information gleaned from websites can't be used thoughtlessly. It must be critiqued.
These are fundamental criteria by which you judge a source of information:
Websites require a second level of scrutiny*:
*From Evaluating Webpages: Techniques To Apply & Questions To Ask, produced by the UC Berkeley Libraries. This is a good, comprehensive guide to evaluating websites.