Your brain can have a break on this one, because shutter speed is much simpler to understand than aperture. A shutter is just a light-proof curtain that covers and uncovers the film or digital sensor inside your camera for a specific period of time. Normally, the shutter curtain is closed, protecting the image sensor from light until you are ready to take a picture. When you press the shutter button, the shutter uncovers the sensor for a certain amount of time and then immediately recovers it. The amount of time that the sensor is uncovered and exposed to light is called the "shutter speed".
Fortunately, shutter speed uses normal, familiar units: the second. Shutter speeds are generally given in fractions of a second rather than decimals (i.e. "1/8", not ".125"). This doesn't mean all shutter speeds are less than a second long, in fact some exposures can take several minutes or even multiple hours in special situations, but most of the time you will be dealing with fractions of a second.
Like aperture, the difference between shutter speed settings is talked about using "stops". A difference of one full "stop" (whether we are talking about aperture or shutter speed) is either double, or half, the amount of light. With shutter speeds, this corresponds directly to either doubling the shutter speed or halving it.
For example, here's a progression of shutter speeds, all one stop apart:
1/15, 1/30, 1/60, 1/125, 1/250, 1/500
The longer the shutter is open, the more light makes it into the camera. The shorter the shutter speed, the less light admitted.
Let's revisit the garden hose analogy. If I turn on the hose for 15 minutes, I'm going to get a certain amount of water. If I leave it on for twice as long, 30 minutes, I'm going to get twice as much water. This may be too much water, it may be too little, but the point is I can control how much water I get by changing how long I leave the hose turned on. The goal is to get the correct amount of water for the container I'm trying to fill, whether that's an Olympic swimming pool or a Dixie cup.
The same is true for photography. You want to get the "right" amount of light to create a correct exposure. Next we'll talk about the final piece of the puzzle, sensitivity (or ISO, or film speed).
PS - shutter speed, like aperture, affects other things besides exposure, such as sharpness and whether moving objects appeared blurred or frozen. We'll get to that later on when we discuss how to use shutter speed creatively.
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