I was practicing guitar — in this case, sight-reading through a book of intermediate/advanced exercises in all different keys — when I found myself, as usual, distracted by reflective ideas about how I was processing the music. In this case, I found myself modulating back-and-forth between processing the notes in an absolute manner versus a relative manner...
Sitting down to write this, I worry about how to take on one of the most controversial issues in all of music in a short article. Entire books and websites and learning programs have been made on this topic. I do not have time or space to review them all or cite all the vast amounts of research, but the Wikipedia articles are a good starting point: Absolute pitch / Relative pitch.
There is more than enough evidence to conclude that all normal people are sensitive to both absolute and relative pitch. Some rare people are highly sensitive to absolute pitch, including the ability to explicitly name any pitch regardless of context — a skill often called "perfect pitch." All my personal acquaintances who have this ability are children of piano teachers or at least started music training at a very young age. It seems that full development of "perfect pitch" requires explicit connection between consistent sounds and consistent names during a critical period in early childhood. Nevertheless, everyone seems somewhat sensitive to absolute pitch (see The Levitin Effect).
Sweet Anticipation, David Huron argues (among a great number of other insights) that absolute pitch is unremarkable (it is easy to understand a neuron firing for particular pitches, and absolute pitch is how most animals recognize sounds). In contrast, the uniquely human ability to recognize relative pitch patterns is more cognitively significant. He supposes that when most children pick up on the idea that songs can follow patterns independent of their exact frequency, sensitivity to absolute pitch is actively suppressed in favor of relative focus.