A bagel is a fairly small round piece of boiled bread with a hole in the center. It is usually either eaten plain or cut in half with some sort of spread in between the two halfs.
The exact origin of the bagel is unknown, but it is said to have originated in Poland. For example, in 1610 in Crakow, Poland, it was said that "beygls" would be given as gifts to women in childbirth. The story that most people accept as the "origin" of the bagel took place in 1683. A Viennese baker wanted to honor King John II Sobreski of Poland ("Vaneusker of the Turks") and thank him for saving Vienna from Turkish invaders. Since the king was renowned for his horsemanship and it is said that the Austrians had clung to the king's stirrups as he rode by after the victory, the baker decided to shape the bread in the shape of a riding stirrup and boil it. In Austria, the word for stirrup is "beugel" and bagel is said was derived from this term. The Yiddish and German word "bugel" means a round loaf of bread and this is also a likely origin for the name.
Bagels remained a breakfast staple in Eastern Europe for the next two centuries and in 1880 when thousands of Jews from that region immigrated to America bagels began to spring up there. Vendors started selling bagels in New York City and Chicago since both cities had large Jewish populations at the time and bagels were still a traditionally Jewish food. When cream cheese first came on the market as a mainstream product in 1920, the New York Jewish community embraced it and it became standard to spread it on bagels.
Up until the 1960s or so, bagels were still considered an ethnic food until a man named Murray Lender came up with the idea to freeze bagels and sell them in supermarkets as a breakfast food. As he put it, his dream was to have a "bagel in every toaster". This brought them into the mainstream American diet and today they're eaten all over the country for all meals as either a breakfast food with some sort of spread or even the recently popularized bagel sandwich.
Generally, bagels are made just of flour, water, yeast, and malt. It's become common to put toppings on them such as poppy seeds, onion, mix in egg, or just about anything else you can think of. Bagels are the only kind of bread that is boiled, as bagels are boiled before baking them. The boiling gelatinizes the gluten in the dough which seals the surface and creates the "shell' that bagels have with a soft center.