This is a stable search term assigned to an article to help retrieve it easily.  It is made up of numbers and letters (example: 10.1234/NP5678) and is assigned by a registration agency (such as CrossRef or DataCite) when the article is registered with them. 

When you enter the DOI on the registration agency's web page search box, it will provide more information on the article, including the latest URL provided by the organization registering the article, taking you directly to the article.   This system was created so that when an organization changed the article's URL, they only had to change it in one place (the registration agency) and people would still be able to find the article.  Prior to this system, when an article's URL changed there was no way for the organization to get the change to everyone trying to use or link to the old URL.  They only had to change it once, with the registration agency and the DOI would remain stable and provide a path back to the current URL.  DOIs are primarily used by the producers of scientific articles.