What is publishing?
Publishing rights are the rights to a song. If you write a song by yourself, you own the publishing (and copyright) from the moment you finish the song. You don't have to set up shop as a music publisher to own those rights - they automatically come with authorship.
There are actually two halves in every publishing dollar. One of the halves is called the "Writer's Share," and that half almost always stays with the writer for life. The other half is called the "Publisher's Share'" and you own that half as well, unless you sign a publishing or co-publishing deal. You can give up some of the publishing rights you own (typically half or all of the "Publisher's Share") in exchange for a cash advance from a music publisher. The publisher will then act as a songplugger to get your song cut by an artist or placed in a movie or TV show. They don't make money unless they get the song used.
Do I need to start my own publishing company?
Typically not. You already own all of your publishing rights from the song's inception. People typically start a publishing company when they get a song "cut" and need a mechanism or company to which they can have the income flow. One word of advice, it's often a better idea to have another company administer your publishing when you have your own publishing company. That means that they will take care of all the business of collecting and disseminating the money that is generated by the song. Many small or individual publishers aren't expert enough to do that on their own.
How do I make money from publishing?
A song generates money for the writer(s) when it appears on an album, gets played on the radio, used in a TV show or movie, gets sold as sheet music, and even when it is used as a ring tone on a cell phone. When the song is part of an album or is sold as sheet music, the writer gets a mechanical royalty. When a song is used in a movie or TV show, a performance royalty is paid to the writer(s).