Authors are always very happy when their paper is accepted for publication in a journal, as this shows that their work was deemed important but editors and referees. But they also want to make sure that their work gets read and does not disappear behind a subscription wall. There are several steps an author can take here.
The author is the copyright holder until this is transfered to someone else. Publishers asks very soon after a paper is accepted for publication that the copyright be transfered to them. Typically, the form asks for all rights, which implies that the author cannot use her own work in other publication or in presentations, even in her own classroom. There are two ways to avoid this: 1) ask for the “other” copyright form, which publishers provides upon request only. This form allows the author to retain certain rights. 2) amend the copyright form. SPARC has developed a standard form that is available here [pdf]. See further details regarding this procedure.
Keep pre-prints online
In many cases, a paper was previously made available online as a working paper. Do not remove it. Indeed, you are the copyright holder and do not have to relinquish this. Even if you did not follow the steps above, in most cases, you can still keep your working paper online. Many publishers have made public that they tolerate, to various degrees, that these pre-prints remain in place. You can check this at SHERPA/RoMEO.
You can even archive so-called post-prints. These are accepted versions on your article. Many universities and research funders actually require that post-prints be publicly archived, for example in an institutional repository. In Economics, it is also common to publish an accepted work in a working paper series. Again, to see what publishers officially allow in this respect, see SHERPA/RoMEO. You have more rights, of course, if you took steps to retain them.