The license you choose defines the social contract you wish to set up among your co-developers and users. The copyright you put on the software will function mainly as a legal assertion of your right to set license terms on the software and derivative works of the software.
Anything that is not public domain has a copyright, possibly more than one. Under the Berne Convention (which has been U.S. law since 1978), the copyright does not have to be explicit. That is, the authors of a work hold copyright even if there is no copyright notice.
Who counts as an author can be very complicated, especially for software that has been worked on by many hands. This is why licenses are important. By setting out the terms under which material can be used, they grant rights to the users that protect them from arbitrary actions by the copyright holders.
In proprietary software, the license terms are designed to protect the copyright. They're a way of granting a few rights to users while reserving as much legal territory is possible for the owner (the copyright holder). The copyright holder is very important, and the license logic so restrictive that the exact technicalities of the license terms are usually unimportant.
In open-source software, the situation is usually the exact opposite; the copyright exists to protect the license. The only rights the copyright holder always keeps are to enforce the license. Otherwise, only a few rights are reserved and most choices pass to the user. In particular, the copyright holder cannot change the terms on a copy you already have. Therefore, in open-source software the copyright holder is almost irrelevant -- but the license terms are very important.
Normally the copyright holder of a project is the current project leader or sponsoring organization. Transfer of the project to a new leader is often signaled by changing the copyright holder. However, this is not a hard and fast rule; many open-source projects have multiple copyright holders, and there is no instance on record of this leading to legal problems.
Some projects choose to assign copyright to the Free Software Foundation, on the theory that it has an interest in defending open source and lawyers available to do it.
For licensing purposes, we can distinguish several different kinds of rights that a license may convey. Rights to copy and redistribute, rights to use, rights to modify for personal use, and rights to redistribute modified copies. A license may restrict or attach conditions to any of these rights.
The Open Source Initiative is the result of a great deal of thought about what makes software ``open source'' or (in older terminology) ``free''. Its constraints on licensing require that:
An unlimited right to copy be granted.
An unlimited right to use be granted.
An unlimited right to modify for personal use be granted.
The guidelines prohibit restrictions on redistribution of modified binaries; this meets the needs of software distributors, who need to be able to ship working code without encumbrance. It allows authors to require that modified sources be redistributed as pristine sources plus patches, thus establishing the author's intentions and an ``audit trail'' of any changes by others.
The OSD is the legal definition of the `OSI Certified Open Source' certification mark, and as good a definition of ``free software'' as anyone has ever come up with. All of the standard licenses (MIT, BSD, Artistic, and GPL/LGPL) meet it (though some, like GPL, have other restrictions which you should understand before choosing it).
Note that licenses which allow noncommercial use only do not qualify as open-source licenses, even if they are decorated with ``GPL'' or some other standard license. They discriminate against particular occupations, persons, and groups. They make life too complicated for CD-ROM distributors and others trying to spread open-source software commercially.