The publisher is the top boss, the person overseeing all apsects of the paper on both the editorial (news) side of things as well as the business side. However, he or she often has little involvement in the day-to-day operations of the newsroom.
The Editor in Chief
The editor in chief is responsible for all aspects of the news operation, including the content of the paper, hiring, budgets, etc. The editor's involvement with the day-to-day running of the newsroom varies with the size of the paper. On small papers, the editor is very involved; on big papers, maybe less so.
The managing editor is the one who directly oversees day-today operations of the newsroom. More than anyone else, perhaps, he or she is the one responsible for getting the paper out everyday, and for ensuring that it's the best that it can be. Again, depending on the size of the paper, the managing editor may have a number of assistant managing editors responsible for specific sections of the paper, such as city news, sports, etc.
Assignment editors are those directly responsible for specific section of the paper, such as city news, business, sports, features, etc. They are the editors who deal directly with reporters; they assign stories, work with reporters on their coverage, suggest angles and ledes, and do the initial editing of reporters' articles.
Copy editors typically gets reporters' stories after they have been given an initial edit by assignment editors. They edit stories and often also write headlines and do layout.