What is Aperture?
In short, aperture is the amount of light that a lens lets in to reach the camera sensor. In the graph at the left, you can see that the lower the number (ie- f 1.4) the larger the opening of the lens is, therefore allowing more light to reach the sensor, while higher numbers ( f 16 ) allow much less light to reach the sensor. The stops along the way ( f 2, f 2.8, f 4 etc) each allow approximately half the light as the previous setting. This is clearly impactful when shooting in low light or bright light. In low light, an f stop of f 2.8 or less will allow your sensor to absorb enough light to get a photo, while in bright sunshine, an f stop of say f 16 will prevent the photo from being overexposed (completely white).
Aperture also affects the depth of field of an image. At a lower f stop of f 2.8, for example, the depth of field will be shallow, meaning that focus will change rapidly to a blur when not on the exact focal point. This is useful in portraits, as it blurs out the background of the photo to focus on the subject. It extremely shallow depth of field, a subjects eyes may be in focus, but their ears are already blurring due to the shallow depth of field achieved through setting an f stop of f 2 or f 1.4. Correspondingly, a faster shutter speed will need to be used at this type of setting, as the amount of light hitting the sensor is so great. This can be a pleasing affect in a portrait, but not so in a landscape.
In a landscape photo, the photographer typically wants to have everything in sharp focus to achieve a more pleasing photo. In this case, most landscape photographers will use a higher number f stop of f 11 to f 16. This creates a depth of field where nearly the entire photo is in perfect focus. However, a tripod will often times need to be used, as to use such an aperture/f stop, the shutter speed will need to be slower, in order to get enough light to the sensor. A slow shutter speed would result in a blurry photo, therefore necessitating the use of a tripod. (Need a good tripod, check out my suggestions here.)
Finally, on the dial of your camera, you will see an A or Av setting. This is the setting where you pick the aperture you want to use, and let the camera set the shutter speed and ISO for you. This can work well under many circumstances, although there will be times when shooting in this mode will not result in the desired affect to your photo. See the article on when to shoot in manual mode for reference.