Topology began in the German town of Konigsberg. Gentlemen of leisure in the cafe society of Konigsberg amused themselves planning a walking tour of downtown Konigsberg that crossed each of the seven bridges in the downtown area once and only once. After years of failed attempts, they wrote a letter to the most famous mathematician of the day: Leonhard Euler. He developed a mathematical proof that such a tour was impossible, his 1736 paper outlining the it is now considered the first published on a topological theme. The problem is the same no matter how long or how far apart the bridges are. It is the interconnection of the bridges that is the essence of the problem.
Topology is sometimes called "rubber sheet geometry," because topology replaces the rigid plane of classical geometry with a rubber sheet. Squares are considered the same as circles, because squares can be continuously transformed into circles without tearing or breaking. Topologist look for more fundamental distinctions -- like holes. One of the classifications of geometrical objects is genus -- the number of holes in the object. Squares and circles are both genus 0 -- no holes. Donuts and coffee cups are identical, as they are both genus 1. A coffee cup with ho handle, however, would be genus 0.
Topologists love strange objects, such as one sided objects or three dimensional objects where the inside and the outside are the same side. Probably the most famous of these is the Mobius strip. You can make one of these by taking a strip of paper and gluing the ends together -- giving the band a half twist before applying the glue. Without the twist it would be a paper band, with a clearly distinguishable inside and outside. The Mobius band, however has only one side. If you put a pencil to the band and move the band until the pencil line reaches the starting point, you will find that the entire surface of the band is marked -- without crossing an edge.
Knot theory is considered a part of topology. The knot known as a bowline is the same if it is tied in a tiny string or in giant ropes a foot in diameter. Size and distance are unimportant -- the bowline is something more fundamental. One of the basic differences between geometry and topology is the transforms that an object can go through and remain the same. In geometry an object can be moved, rotated and flipped over and remained unchanged. In topology it can also be stretched and bent, but not torn.