What is the Redshift?
Imagine you are out for your morning walk when a giant bus comes barreling down the road toward you. As it approaches, the driver lays on his horn and you jump out of the way. The closer the bus gets, the higher the pitch of the horn. As the bus passes, the pitch of the horn becomes lower. This is change in sound as something approaches and recedes is called the Doppler Shift. As the bus moves closer to you, the wavelength of the sound waves is shortening and the frequency of the waves is increasing. As it moves away, the opposite happens: wavelength is increasing and the frequency is decreasing.
Light also moves in waves (as well as particles--but I digress). The same process that we have observed with sound also happens with light. When an object is emitting light and moving toward you, the light wavelength decreases while the frequency increases, in other words, it gets compressed (or squished). This wavelength corresponds to the blue end of the visual color spectrum. This object is said to be blueshifted. Inversely, when an object is moving away, the wavelength of the light it is emitting increases, while the frequency decreases: it gets stretched out. This corresponds to the red end of the visual color spectrum, and the object is said to be redshifted.
In the 1920's Edwin Hubble was using the concept of redshift and blueshift to calculate the distances of "nebulae" (later known as galaxies). He found that all of the galaxies outside our local group were moving away from each other. For centuries, the Universe had been a static place, and now we knew that it was not only moving, it was expanding! The newfound knowledge of the expanding Universe led directly to the theory of the Big Bang.
In 1917, Albert Einstein had found that his newly developed theory of general relativity indicated that the universe must be either expanding or contracting. Unable to believe what his own equations were telling him, Einstein introduced a cosmological constant (a "fudge factor") to the equations to avoid this "problem". When Einstein heard of Hubble's discovery, he said that changing his equations was "the biggest blunder of [his] life."