Unlike periodicals and journals (see introduction to "Article Indexes" on the "Periodicals" page), few newspapers were ever indexed. Before newspaper digitization, the main way to find newspaper articles was to sit at a microfilm machine, and browse through issue after issue of newspapers. If you were researching a specific event, and had a date and place associated with that event, then your work would be a little easier.
The few newspapers that were indexed (e.g. the New York Times) were among the very first to be digitized, so with very few exceptions, you will not be using indexes to identify articles from newspapers.
To find a specific newspaper, or to identify newspapers published in a specific year or place, or newspapers published for a specific audience (e.g. African Americans, farmers), use the:
Many newspapers have been digitized, and the University of Illinois Library has a strong collection of digitized newspapers. Below are the best databases to begin with, when researching nineteenth century American history.