If you had to boil it down, what is respiratory failure?

Several reasons. Respiratory failure usually occurs due to one or more of the following...The inability to get oxygen into the blood, the inability to get carbon dioxide out of the blood, a problem with the nerves or muscles that do the work of breathing, or airway obstruction.
Lung failure.., Respiratory failure occurs when the lungs are not able to function normally due to some underlying etiology. It results in a high carbon dioxide level and a low oxygen level in arterial blood. Treatment may include mechanical ventilation if severe enough as well as treating the underlying cause of the failure. Many etiologies can lead to respiratory failure.
Lungs dont work. The respiratory system is made up of the lungs and the breathing muscles including the large breathing muscle the diaphragm separating the chest and abdomen. Its job is to allow the lungs to inflate with air so oxygen and be obtained and carbon dioxide, your bodies waste gas expelled. When something damages this system, to the point it doesn't work you have respiratory failure.