To get a job as a copy editor, you must have a rock-solid sense of grammar, spelling and punctuation and a keen eye for errors and typos. You must be able to read quickly and accurately. You must be able to write well and speak well. You must be able to work alone with very little direction, but you must also be able to work as a team member under a micromanaging senior editor; either work scenario is possible, as is anything in between.
Copy editors at magazines and book publishing companies check materials for grammar, punctuation, readability, style, etc. They also generally do fact-checking and suggest minor revisions. They may do research for writers, and they may be called upon to produce materials for websites.
People seeking entry-level copy editing positions at magazine/book publishers generally need a bachelor's degree in English or journalism. However, publishers that produce scientific, technical, or highly academic works will often accept (or might even require) degrees in relevant academic fields along with evidence of being able to do the particulars of the editing work.
Editing for technical publications often pays better than similar jobs at mainstream publications. So, if you want a job as an editor and are majoring in something besides English or journalism, you can do well provided you get some decent experience on a student paper or magazine while you're in school. Conversely, English and j-school majors will do well to supplement their degrees and student editing experience with a good grounding in other disciplines, particularly the sciences.
Copy editors on newspapers are usually called upon to create the headlines for stories in addition to editing copy; they will also often lay out the stories.
I worked as a copy editor for a small daily paper many moons ago when I was in grad school. It was a tedious, stressful, utterly thankless job. The paper had to be laid out and delivered to the printer a little before midnight, and often reporters and desk editors didn't get copy delivered until well into the evening. I remember several instances in which there were two of us who had less than an hour to check the entire paper before it had to be sent to press.
Newspapers often have a hard time retaining copy editors, which doesn't surprise me given my own experience. The pay was practically nil, you got little appreciation from the other staff when you did your job properly, but if you messed up and overlooked something, you got your butt chewed out.
The only advantage to working as a newspaper copy editor is that it gives you invaluable experience so that you can get better, saner work later in your career. If an employer sees that you could cope with daily newspaper work, he or she will know you can handle an enormous amount of deadline stress and chaos.