What It Means to be Published
Publication is very straightforward: it means sharing your writing with someone else. That someone can be your grandfather, your writing group, or the readers of The Atlantic Monthly. If you send your poem to twelve magazines and get twelve rejection slips, you can at least be sure that twelve people have now read that poem, that it has entered their consciousness and resides there, somewhere, which is, really, one of the reasons we write (and publish): to enter the consciousness of someone else. The same reason we speak. To be heard, to be noticed, to change, however minutely, the world around us.
That said, there are a number of different types of and venues for publication. All types of publication, except self-publication, indicate that someone has selected or screened your work. While you may not actually get money, it's proof that at least one person has deemed your writing worth the time it takes to read it: thus, more people (especially editors) will be willing to consider reading other writing of yours. In other words, getting published gives your writing credibility.