The exposure triangle is crucial in making sure exposure is correct, so the image does not turn out under or over exposed; if one aspect of the triangle is not on the right setting in correlation to the others, the exposure will not be correct.

This is demonstrated in this diagram:
There are many different camera modes which are optimized for different projects, but the basic ones are:
APERTURE VALUE (AV)
AV prioritises aperture over shutter and film speed. This means that when the aperture is changed, shutter and film speed will automatically change to make the exposure correct.
TIME VALUE (TV)
TV prioritises shutter speed in the same way AV prioritises aperture.
MANUAL
Manual mode means the user must change all of the exposure triangle manually to achieve the correct exposure.