The first, and most obvious, place to start your research is in your library's reference section. There you'll discover a variety of biographical indexes with short histories of persons in occupations such as authors, musicians, scientists, educators, and politicians. Even if the person you're researching didn't do anything to make them famous, they may still be well known in their field. Starting here will offer you the chance to at least get a brief history of your ancestor.
The next place to look, believe it or not, is in the children's books section. Besides all the fiction books on the shelves, you'll find quite a few non-fiction books about various occupations and some encyclopedia-style books on peoples of the world. The Peoples of North America series, a collection of books that covers 52 different immigrant groups, is a great place to begin. These books explain when the immigrants to America left their homeland, where they went when they arrived in North America, and why they settled here. At the back of each book is a list of sources which you can find elsewhere in your library. Another one of these compendium book series is one on America's wars, which discusses the fighting of wars in the battlefields and on the home-front. This will be invaluable if your ancestor was ever in the military.
After delving into books in the children's section, head on over to the map department. If you're looking for the location of someone who lived in the mid 19th century, you should look at a map from that time period. Since towns and counties change jurisdictional boundaries–many merged to form new municipalities– don't assume the present location of a town is the same as it was back then. Old maps can provide you with lots of information about county and town names and boundary changes. Some maps may even show where a particular family lived, so you know what part of the country you need to focus on.
If you need a break from heavy research, wander over to your library's fiction section. Choose a novel that's based on or written in the time period and country of your ancestor. Many of today's historical novelists do extensive research before writing their books. Take mystery author Elena Santangelo (Link: http://www.elenasantangelo.com), who brings the past alive through her book series' main character, Pat Montella. By reading one of her books, you'll perhaps get a feel for what your ancestor lived through and can interject some social history into your research. Some authors even include references to their historical sources at the back of their books.
Another important source that you may not have thought of is microfilms, stored on reels, and microfiches, stored as flat sheets. Perhaps you're hesitant about using the microfilm and microfiche readers because you don't understand how the machines work or think you can find the same information elsewhere. Remember, libraries have to house a lot of material, so space is limited. They often store many of their reference directories on microfilm. You can make copies of anything you find on microfilm, even though you can't copy the same directories on the shelves.
You'll also find old city directories and telephone books on microfiche. You'll discover these are a valuable resource in searching for where your deceased relative lived or in locating living missing ones. Since city directories go as far back as the 18th century and if you know your ancestor's country of origin, you can use them to locate someone fast.
Libraries also store old copies of newspapers on microfilm and microfiche. If you know the date and location of an event, you can check the old local papers. Your ancestor may have had an article written about them if they were active in the community or if their business had been profiled. Even if they weren't well known, you may find a mention of them in a related article. Take some time to learn how to use this valuable tool. You'll be glad you did.
Some libraries have extensive photographic collections. Others house collections of letters, journals, diaries, as well as family photo albums. Most of these collections are local. However, if you're traveling to the area your relative lived, it may be worth looking through them at the local library. Even if your ancestor didn't live in that area, they may have passed through it and left some record of their visit.
Lastly, many modern libraries have computers accessible to the public. While these are convenient for those who don't have a computer or Internet access at home, they'll be just as useful even if you do. One important online source is Family Search, from the Church of Jesus Christ of Latter Day Saints. It consists of the International Genealogical Index (IGI), a Family History Library catalog, Ancestral File, and U.S. social security and military indexes.
At state, regional, and county libraries, you can also access the card catalogs of other libraries within the system. This is particularly handy if you know of a book you need for your research but cannot find it in your library. Once you have located the book, you can request it through interlibrary loan.
The key to finding what you need is to use your imagination. Be creative in your approach. Just because information isn't where you think it should be doesn't mean it doesn't exist somewhere else.