Aperture simply means "opening", and in photography we use the term to refer to the diameter of the opening in a special adjustable diaphragm within each lens. When this diaphragm is constricted, there's less physical space for light to get in, so naturally the exposure is darker, and a more-open aperture allows more light and results in a lighter exposure.
Aperture has another important property. When the aperture is very small, the admitted light is highly "collimated", which is a fancy way of saying "all the rays are nicely parallel to each other". This results in a sharp focus for all the light that comes in. When the aperture is more open, only the rays which closely match the focus point are collimated — which means that whatever you've focused on is sharp, but farther or closer parts of the scene will be increasingly blurry.
Lens aperture is usually given as an f-number, which is a ratio of the focal length of the lens to the effective size of the aperture. This is more convenient than using physical diameter, because it works out that the actual amount of light gathered for a given physical aperture size depends on the focal length — so, if one uses the ratios, the exposure is the same regardless of lens length. (The counter-intuitive side-effect of this scheme is that smaller f-numbers let in more light.)
These f-numbers are used in photography in a sequence which may seem irrational: f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, and so on. However, these are chosen for a simple reason: each one has twice the area of the previous, letting in twice as much light. (There's no mystery to the numbers — the area of a circle is π × radius squared, and you can quickly work out that to double the area, one must simply increase the diameter by a factor of √2.)
Each step in the sequence is called "one stop", presumably because on manual lenses there's a physical catch which makes the dial which controls aperture nicely stop at each of these pre-defined points. However, the term "one stop" is generally also used by analogy for shutter speed and sensor ISO to mean "the amount by which this factor must be adjusted to double or half the recorded exposure".
An important thing to keep in mind is that while a small aperture excludes non-parallel light rays, as the size of the aperture approaches the wavelength of the light being captured, another effect comes into play: diffraction, which is a bending and scattering of waves as they pass through the aperture. In practical terms, this begins to affect APS-C form-factor cameras at around f/8, and so stopping down much beyond may increase depth of field at the expense of decreased sharpness in the in-focus areas. At some point, the effect of diffaction becomes strong enough that stopping down the aperture further doesn't provide any benefit at all.
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