Academic credit is based on publications. This is not news to anyone, of course. Most papers (in biomedicine) have more than one author, which means that at some point there has to be a decision on what order to put the authors in. There is a standard sort of rule (again, in biomedical-type fields, other fields have their own conventions):
First author - the student/fellow/trainee that did most of the work.
Middle author(s) - people who helped out with experiments or contributed unpublished reagents, etc.
Last author - the PI of the group. In general, the person that you direct correspondence to regarding the paper.
For us biomed types, you get the most "credit" for a paper if you are the First author or the Last author. First author credit gets you a good postdoc, maybe a PD fellowship, and hopefully a job. Last authorship gets you tenure and grants. Middle authors get ... a pat on the back, a high-five for collaborating/playing well with others, a line on your CV (which can be a big deal).
Sometimes it is not that easy to decide what order the authors will be listed. Maybe two labs collaborated and one grad student from each group did a lot of work. Maybe a new postdoc picked up and finished up a project that had been started by a different lab member. Maybe you have a conjoined twin. Whatever. Discussions and wrangling about order of authors can get nasty, because we all recognize the importance of being First or Last author. Some folks have tried to get around this by designating Co-First authors. Maybe this placates someone who thinks they should get more credit. That's crap, as DrugMonkey and CPP, and probably a million others have said before.
The problem is, this is bullshit because there can only be ONE First first author. The second co-first author is NOT first. This came up on twitter today when I noticed a tweep proposing SWITCHING the order of co-first authors on a CV. This is a BIG NO-NO!! Yes, I understand that if the co-first authors were really equal contributors that it wouldn't matter. But that is not the reality. And if I read your CV and look you up on Pubmed and the author list has been changed, then I'm going to look at your application like CPP - with extreme prejudice. Maybe not everyone has the same view...but do you want to take the chance that your CV gets trashed? Don't do it.