What is Exposure

Exposure is the amount of light hitting your Camera's sensor.

It is typically measured in units of "EV" and typically expressed as a combination of Shutter speed, Aperture and Film Speed that were used to achieve a specific value of EV.
You are typically trying to achieve EV = 0 which indicates unbiased exposure for a given area of your composition.
Negative EV values indicate "underexposure" whereas positive EV values indicate "overexposure"

How is exposure determined?

Your camera has a light meter. This measures the brightness of the scene and determines the amount of light the camera needs to properly expose the scene. The light meter can tweak the light needed based on various factors like exposing just a spot ("Spot Metering") of the scene or the entire scene ("Average") or somewhere in-between ("Evaluative").

How can you control Exposure

You use the following tools to control the exposure. In a sense Exposure is the recipe with the following key ingredients.

Shutter Speed
Aperture
Film Speed (ISO)
Ambient Light

Shutter Speed

Shutter speed is the length of time a sensor is exposed to light. It is controlled by opening the shutter for a brief amount of time.
Shutter speed is expressed in fractions of seconds when it is less than a second, otherwise it is expressed in seconds.
Typical Shutter speed values are: 1", 1/2, 1/4, 1/8, 1/15, 1/30, 1/60, 1/125, 1/250, 1/500, 1/1000 and so on.
Doubling the shutter speed halves the light or halving the shutter speed doubles the light entering your camera.
Increasing shutter speed reduces the light entering your camera.
Changing shutter speed typically does not change your composition unless you are going below 1/60th of a second.

Aperture

Aperture is the size of the hole in your lens for letting the letting in.
Aperture is measured in "f-stops".
Aperture values may be listed as reciprocals so f/1.4 instead of 1.4f or f/5.6f instead of 5.6f. The value remains the same.
Aperture is listed as a series of numbers typically ranging from 1.4 to 32 with each number being about twice the number two positions to the left of it
Typical f-stop values are: 1.4f, 2f, 2.8f, 4f, 5.6f, 8f, 11f, 16f, 22f, 32f and so on
Doubling the aperture halves the light or halving the aperture doubles the light entering your camera.
Increasing aperture reduces the light entering your camera.
Changing aperture changes your composition as it impacts the depth-of-field of your composition.

Film Speed

Film Sped is the measure of the sensitivity of your camera's sensor to light.
Film Speed is measured in ISO units.
Film Speed values are listed as numbers where every number is double of the number on the left.
Typical Film Speed values are : ISO 100, ISO 200, ISO 400, ISO 800, ISO 1600, ISO 3200, ISO 6400 and so on.
Doubling the film speed effectively doubles the amount of light in your photo.
Decreasing ISO decrease the amount of light.
Changing Film Speed changes your composition as it impacts the level of noise in your composition.

Ambient Light

Ambient light is the natural or artificial light available physically where a photo is being taken.
Ambient light can have a big impact on exposure of your photo.
You use Shutter speed, Aperture and Film Speed to make the most of your ambient light or to make up for the deficiencies of natural light in a given scene.

Combining Shutter speed, Aperture and Film Speed

Achieving an ideal exposure is the practice of combining Shutter Speed, Aperture and Film Speed to compose a photo that satisfies your aesthetic judgement for a given scene.

Mixing and matching various values of Shutter Speed, Aperture and Film Speed could produce the same exposure but different visual effects.

Like three different faucets filling a water tank - different settings for each faucet could fill the tank with the same amount of water. However, unlike a water tank - the settings of faucets could make the water look and feel different.

for example: EV = f/8, 1/60, 200 == f/5.6,1/125, 200 = f/5.6,1/60,400

Shooting Modes

Aperture Priority. In this mode you set the Aperture and the camera automatically sets the shutter speed and (often) the film speed in order to achieve the desired exposure.
Shutter Priority. In this mode you set the Shutter speed and the camera automatically sets the aperture and the film speed.
Manual. You set everything both the Aperture and the Shutter speed.
Auto ISO. ISO is set automatically by the camera. You can use this even with the Manual mode.

Exposure Tips

Auto-exposure is fine in most cases. Use manual exposure for practice or while using an external flash or when you are going for specific settings.
Pre-Programmed mode (portrait, landscape, night mode) should be avoided.
Aperture Priority is generally your best shooting mode.
Auto ISO is almost always your best option.
Use Evaluative metering for general results.
Given an option between under exposure and over exposure always go for under exposure. Underexposure is easier to correct in post-processing.