You earn a copyright in your scholarly writing or other project as soon as you create it. This includes the right to:
make copies of your work
make derivatives based on your work
distribute copies of your work to the public
publicly perform and display your work
in case of sound recordings, perform your work publicly by means of a digital audio transmission
As the copyright owner, you and any coauthors have the sole discretion as to how and when to use these rights. However, most traditional, pay-to-read scholarly publishers require you to sign over all of these rights to them, meaning you no longer own the copyright in your work.
Some publishers allow you to use your work in certain ways, such as using it in your classroom or sharing it individually with other researchers.
Others will allow you to post a version of your article - usually the one you submitted or that was accepted by the journal - on your personal website or in a digital repository.
Others will not grant you any rights.
This is why it's important to read your contract before signing it so you can understand what rights you will actually have in regards to your article. This is especially important in order to make sure you are allowed to post your article online without violating copyright law.
If the terms of your contract do not allow for sharing versions of your article, you can negotiate for different terms. SPARC and Creative Commons both created sample contract addendums you can use to help you in negotiations. You can also read about one researcher's successful quest to negotiate his license in order to make his article open.
Creative Commons is a not-for-profit group that created a series of six licenses that allow copyright owners to pick and choose which rights they want to grant to the public in their works. Creative Commons licenses work with copyright law and actually give copyright owners certain protections - the right to attribution - that copyright law generally does not.
Although each Creative Commons license includes legal language behind it, they're also represented by easy-to-read icons that include a mix of these symbols:
BY - Attribution. Anyone who uses your work must give you attribution in the way you request but not in such a way to suggest endorsement.
NC - Non-commercial. People are allowed to copy, distribute, display, perform, and modify your work in any way except for commercial profit.
SA - Share alike. People are allowed to copy, distribute, display, perform, and modify your work so long as they distribute any modified work on the same terms you've licensed your work under.
ND - No derivatives. People are allowed to copy, distribute, display, and perform your work but are not allowed to modify it.
Information adapted from Creative Commons.