On a given publication, how do you know who is first author, second author, etc? Is the first author simply the one who is named first?

Also, can there be multiple first authors or second authors?

Yes, the first author is listed first, and so on. Usually the first author is the author that did the majority of the work (in terms of data contributions - they are also usually the person who wrote the paper, at least where I work), the second author contributed the second most, etc. If two people contributed an equal amount, obviously one will have to be listed ahead of the other, but whoever submits the article can choose to indicate that those two people contributed an equal amount (using an asterisk and a footnote or something).

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First author = did most of the work, usually the one who wrote the paper
Last author = usually the PI whose grant paid for the work
Everyone in-between = other people who worked on the paper, often in descending order of contribution

Sometimes the first couple of authors will have a star by their names with a footnote that says "these authors contributed equally to this work." In those cases, it's advantageous to have a last name that starts with a low letter of the alphabet, because most people don't notice the footnote, and it doesn't show up in the bibliographic listing of the article.

Pretty much the first author is one who did most of work and might';ve contributed to writing the paper. The most important person though is a corresponding author nevertheless. Having your name in in between first and last author does not really do you any good or bad, except of course if you got your name in Science or Nature... impact factor of 36...

What do you guys think "the rules" are when there are just two authors? Curious, 'cause I'm trying to get into that situation.

The very broad general rule is 1st author did much of the work operationally, the oversight of the project, and even the writing of the paper. This is usually the person that takes a project from start to finish. Someone senior usually tells them "It would be great to do a project on blah blah blah. Why don't you go and tackle it" to which they actually get it done. They certainly may get some help in the form of research assistants/associates, etc, but they "own" the project and take a lead on it. First authorship is considered a significant achievement, generally.

The last author, also called the senior author, is PI or head of the research group (formal or informal Primary Investigator, head of the lab, Chief of a medical unit, or Director of a certain program, etc). This is the "someone senior" typically from above who has mentored or supervised the project. Their credibility and reputation as a senior author is also factored into each paper they write as last (senior) author.

People in the middle are generally listed in order of contribution from most to least - but this convention is often broken. Also, in clinical medicine, there are definitely a lot of people who only glance at the people in-between (eg. the difference between 2nd and 9th in a 10 author paper is negligible). This rule is broken - in my mind - for more junior scientists (undergrads, pre-meds, research assistants looking to get into med school) because quite frankly getting to contribute to any study in any capacity is a differentiator to those that have no publications of any kind.

If there are two equal collaborators, it's ultimately a flip of a coin. If you are a researcher in someone's lab and only two of you are on a paper, it's almost always: researcher, head of lab. If person A did proportionally more intellectual work than person b, then generally person A goes first, person b second.

This is all convention. Generally, your mentor/preceptor should be able to help guide you as far as author order and giving credit where credit is due.

On our end, they taught us:

1st Author: Person who did most of the work, like write the manuscript, and do most if not all of the experiment.

2nd Author: Person who helped out the most, and/or person who mentored the 1st author (e.g.: if 1st author was grad student) the most.

Then you got every one after that who contributed, e.g: contributing authors.

Last author as some has stated is the guy that got the whole thing started. Usually the PI, who got the grants, and so forth.

First Author: Person sleeping with PI or spouse of PI who did less than one quarter of the work.

Second Author: Person that actually did a vast majority of work who now can't get a job due to a lack of first author papers.

Third Author: Person that started the project and gave up.

Last Arthor: Principle Investigator. (See first author.)