Light enters your camera through the aperture, which can be larger or smaller to control how much light comes in the camera and falls on the sensor. The shutter controls how long that light falls on the sensor.
("DSLR" by Jinho Jung is licensed under CC BY-SA 2.0)
The aperture is the opening in your lens. Basically, it's a hole that lets light in. Most lenses allow you to change the aperture size. Aperture size is measured in f-stops. Oddly, the smaller the f-stop, the larger the aperture or lens opening.
Aperture affects the brightness of your image. The larger the aperture, the brighter the image. The smaller the aperture, the darker the image.
Aperture affects depth of field, which basically refers to how far back objects in your photo appear sharp. Smaller apertures keep more of your photograph sharp, while larger ones keep a smaller area in focus and leave the rest blurry.
("Photo-Basics-Aperture-Infographic" by Esmer Olvera is licensed under CC BY-ND 2.0)
The shutter exposes your camera's sensor to light. Shutter speed refers to the length of time the shutter stays open while taking a picture. Faster shutter speeds expose the sensor to light more briefly than slow shutter speeds. Slower shutter speeds allow light to linger longer on the sensor.
For example, you might use a fast shutter speed to capture an object in motion clearly. It can allow you to "stop" motion because the shutter allows the light in for such a brief time that you can isolate a moving object. A slower shutter speed will show motion through blur, since the object is being exposed for a longer amount of time and is moving.
Shutter speed also affects the brightness of the image. Fast shutter speed means less time for the camera sensor to detect light. Slow shutter speed means more time for the sensor to detect light.
ISO refers to your camera's sensitivity to light. In bright conditions, a low ISO is sufficient. In low light conditions, you may want to use a higher ISO. Higher ISOs can mean more digital noise in your picture, so beware.