What are the main functions on a DSLR camera that affect the correct the exposure of an image?

If a DSLR camera is not set to 'Automatic' mode, there are 3 primary functions that need to be considered when deciding on the correct exposure for your image.
1) the Aperture - how wide the aperture on the lens is set determines how much light can pass through the aperture. A small aperture (e.g. f/16) allows less light to reach the sensor as opposed to a wide aperture (e.g f/2.4).
2) the Shutter Speed - how long the shutter is allowed to remain open after you depress the release on the camera. A short exposure time (e.g. 1/1000 sec) will allow less light to hit the cameras sensor whilst a long exposure (e.g. 1/4 sec) will let more light hit the sensor
3) the ISO setting - this setting is used to control how sensitive the sensor is to light hitting it. A low ISO setting (e.g. ISO 200) is much less sensitive than a high ISO setting (e.g. ISO 1600) which means less light is required to capture the for the image on the sensor.
All 3 of these functions work in tandem to ensure the correct exposure value is created based on the image and effect you are trying to create.