There are some terms you'll hear over and over again in academia, and two of the most common are "paper" and "publish". Papers are the documents you write up to describe your research—usually several experiments all on the same theme. Papers have several parts: an Abstract, Introduction, Methods section, Results section, and Discussion section. In the Abstract, you have a summary of everything you cover in your entire paper (they are extremely useful). In the Introduction section, you describe where your study sits in the "literature", which is the collection of other peoples' papers that have been published on the topic. More specifically, you overview the literature and describe why your study is new and adds to the literature. In the Methods section, you describe what you did. Theoretically, your description will be detailed enough that someone else can replicate your experiment. This doesn't often happen in practice; rather, people contact the original researchers if they want to replicate experiments. In the Results section, you have all of your graphs (which are called "figures"), tables (large sets of numbers— otherwise you put the numbers in the text of the results section), and the text describing the figures (and tables if you have them.) In the Discussion section, you describe what you results mean, and how it relates to the broader literature. Sometimes there's a conclusions section where everything you've done is summarized.
You're going to be reading a LOT of papers. Papers are very hard work to read mostly because they are aimed at people in the field. That means they will rely on concepts that you will likely be unfamiliar with and they will use jargon. Also, in science we tend to write in the passive voice, and there is a wide range of writing abilities, so things can be confusing even if the subject matter wasn't difficult. Usually when I need to learn about a new topic, I start with a review paper— which is a paper that reviews a whole bunch of other papers in the literature. I read that fully and carefully, taking note of which other papers are referenced. Then I go read some more review papers and some of the primary literature ("primary literature" seems to be a somewhat ambiguous term in my head, but I feel it refers to all published studies that aren't review papers). As I become more familiar with the concepts, I start skimming. This differs based on the person, but I usually read the abstract, part of the introduction (usually the end of it, when they start describing the study), and the conclusion. Then I'll take a look at their figures to make sure the conclusions are supported, and skim through the methods if necessary. This is usually what you need to be familiar with if you're referencing a paper—where their study fits into the literature, what point they made, and is their point supported. For papers where I'm not trying to make a strong point, I'll just check the abstract and some of the figures. For papers that I'm using extensively, I'll read through the whole thing and carefully evaluate it.
(I like this post for a description of what literature review looks like for a more experienced scientist.)
Papers are the end-all-be-all of science. They show that you're doing work; they're the main determinants of whether you get funding or not. You'll hear "publish or perish", a colloquial phrase describing the fact that if you don't publish your work you'll be out of a job. Papers can be of differing qualities, and this will be reflected by the journal they are published in. Journals are rated by "impact factor" among other things, which is a measure of how much other people cite the papers they publish. The most prestigious journals vary by field, but everyone wants a paper published in Nature or Science. Another measure of how important a paper is the number of citations it has; usually the ranking of the journal it's published in and the number of citations are complementary in describing how important a paper is.
What's an important paper? Papers that have shown something new and have the potential to drive a lot of other new research. Bad papers are papers which are variants on old research, or papers that don't work. People often don't publish work that didn't do what it was supposed do, though there is a movement trying to push against this. The argument is that people should publish their "negative results" (things that didn't work) so that other people don't try to replicate the work as well, wasting everyone's time.
Where do you find papers? There are different sources for finding papers, like PubMed or Web of Science, which are databases of sorted papers. I use Google Scholar almost exclusively, or the search database associated with my school. There are a lot of great buttons on Google Scholar (Import into Endnote for citations, Related Articles, Cited By) but hopefully I'll get into that in another post.
Are papers free? Papers are not free, but as I was told, you should never pay for a paper. Colleges buy subscriptions to journals so that their students can access papers for free. The nice thing about Google Scholar is that if you have a gmail account associated with a school, the free links to those papers will pop up as you're searching. Otherwise you can look through your school's database. Usually papers are available when you search them. Sometimes they're not—in that case, you can sometimes check the author's website and the paper will be available there. You can also order books from the library at your school if they aren't online, which occasionally works. The problem comes if you aren't associated with a college, in which case you're stuck. The system is unfair and flawed in that people who are associated with specific schools have increased access to papers; there's been a movement centering around "Open Access" that seeks to remedy this, but it's still a solution in progress. In these cases people often check if friends have access.
What are "citations"? Papers have authors. And the funny thing is that who the authors are, and the order of the authors, really matters. Who the authors are matters because every lab has a specific bent—they'll have a set of topics they investigate and a bias in what they're expecting to find. Some labs also do better science than others. Oddly enough, science is just as much about the people as it is about the science, and this is one of the ways that this comes through. The order of the authors on a paper matters. This differs by field. In neuroscience and psychology, the last author is the name of the PI (principal investigator) of the lab—i.e. the head of the lab, the person who's in charge. The PI is consistent across years, and is who has control over a lab's focus. The first author will be the graduate student or post-doc who did almost all of the work: at least collecting the data and analyzing the data. (The PI and graduate student / postdoc will have worked together to design the project and sometimes to write the paper.) Everyone else in the middle are people who did work on the project but weren't the main author.
During a PhD, it's great if you can publish around one paper per year once you get going. This metric varies widely based on field and year. Computational papers are much easier to get out than papers which require an immense amount of data. The last two years of a PhD make up almost all of a PhD student's productivity. And post-docs are highly trained, so they have higher output.
Papers are extremely important in academia, and there are many resources on how to write them. If you're in an undergraduate science career, you'll undoubtedly be trained to read papers, as soon as you transition out of reading textbooks into reading about all of the new stuff that's being done!