DOI stand for "Digital Object Identifier". Now, don't you feel smart?
A DOI is a unique numbering system managed by the International DOI Foundation. If you're really curious you can go to the foundation website and learn all the details.
Basically, the idea is that by assigning digital objects a unique number that never changes it will be easier to find these objects on the Internet. It is similar to an ISBN number for a book that is published and an ISSN number for a periodical.
WHY YOU SHOULD CARE
When the American Psychological Association published its 6th edition style manual in 2009 it began requiring the use of DOI's at the end of the citation for any article published in digital format. If you are using a library database and click on the "cite" tool and view the APA style format you will see that it appears at the end of the citation..sometimes. Sometimes it won't be there.
WHAT IF I USE AN ARTICLE AND IT DOESN'T HAVE A DOI?
If you accessed the article in its print form off of a library shelf, look for the DOI on the first page of the article near the top or bottom edge of the page (usually). If it's provided - use it. If you can't find one, don't worry too much about it. However, if you want to make sure you can always type the title of the article into a Crossref.org search and see if there is one. If so, use it. If not - obviously, you can't include it, but at least you tried.
If you accessed the article in its digital format you should always try to include it, if you can find it. There is a tool for finding DOI's at Crossref.org Simply enter the author's name and the title of the article. If your results come back with "no DOI", just cite the article without one.
Since APA's requirement is rather recent, as well as the DOI system, we are in a period of transition. Articles published before 2009 may not have one. Therefore - if you have a DOI, include it in the citation. If you don't have a DOI and can't find one using the DOI look-up at Crossref.org, then you might wish to include the name of the database and/or website URL for the journal.
Always check with your professor and see if he or she has a preference.
If the library database or journal publisher's website provides no DOI it can either be searched using CrossRef.
The DOI system was implemented in 2000, so if an article precedes that date it is possible it has not had a doi assigned to it. Just as in other elements of a reference - if you do not have the information, you simply skip it and craft the citation without it.