Citations are necessary for a number of reasons. Not only is it expected by most of your professors, but it is the way that scholarship and ideas progress, by giving credit (and thus incentive) to the authors of the original ideas. Additionally, using citations:
Think about this: how would you prefer your own ideas to be used by others?
Just because it's on the Internet, does not mean it is free to use or "in the Public Domain". In fact, almost every image, video, and sound clip on the Internet is copyrighted by default in the United States (regardless of whether it actually says "copyrighted" anywhere).
Even if your assignment doesn't require you to cite multimedia content, it's good practice to include the author's name and the source of the image, video, or sound file (e.g. the name of the website or the URL). If your professor requires that your citations be in a specific format, such as MLA, APA, or Chicago, you should know that there is a standardized way to cite almost anything you find online: webpages, videos, tweets, games, you name it!
DOI, or "direct object identifier" is a way to track articles and electronic documents on the web. Whereas the URL of a journal may change over time, the DOI for each article always stays the same.
Crossref.org is a resource for looking up DOIs based on limited citation information and, reversely, for looking up citation information based on a known DOI.
A "citation" is the way you tell your readers that certain material in your work came from another source. It also gives your readers the information necessary to find that source again, including: the author of the work, the title, the name and location of the company that published your copy of the source,the date your copy was published, the page numbers of the material you are borrowing.
This research guide will give you a brief overview of the most common citation styles and point you in the direction of more detailed resources and useful tools.
(image source: xkcd, used with permission CC by-nc)
The general rule of thumb for citations is "If it's not common knowledge and it's not your own idea, you need to cite it." Since you know your own mind, perhaps a better question than "when should I cite" is "what is considered common knowledge?"
According to the Purdue Owl website, information is common knowledge if:
You can also use the "man on the street" rule: if you can walk up to a group of people on the street and at least one of them knows the information you need to cite, then it is probably common knowledge.
However, what is common knowledge can vary within the context of a specific class or a field. For example, what is common knowledge to mathematicians may not be common knowledge to psychologists. This is why, for example, a mathematics article will not cite that Carl Gauss coined the idea that mathematics is the "Queen of the Sciences," but a psychology article might.
(image source: shishberg on flickr, cc by-sa)