The problem in offering a discrete answer to this question is that the concept of friction covers a wide variety of phenomena at various macroscopic and microscopic levels (down to the level of electrochemical bonds) that may or may not be interrelated and thus, no one explanation is going to suffice to describe all surface friction phenomena. To compound this, the term “friction” is also sometimes used to describe internal continuum mechanical losses (hysteresis) in solids and Newtonian fluids and lossy behavior in aeroelastic, turbulant conditions with non-Newtonian fluids. I consider this use of the term friction to be inappropriate, or at least highly confusing, but it is frequently use in technical jargon and therefore an accepted use of the term.
If you want a simple answer for solid surface-to-surface static friction can be broken into two essential domains; that caused by surface roughness which results in normal forces between “interlocking” surface features (think of dragging two pieces of sandpaper against one another), and the tangential forces that result from having to break attractive electrical bonds between two surfaces, or between a surface, a adhesive medium, and another surface (like two plates of glass with a thin layer of water in between them). When you get to dynamic friction things get far more complex; surfaces that had low friction statically can suddenly have high friction dynamically (particularly in the solid-fluid-solid case where the internal viscosity of the fluid suddenly dominates the effect), and conversely, rough surfaces with high static friction may have much lower dynamic friction because of the intermittant contact permitted by surface roughness. In addition, the basic character of the interface may change; for instance, in a car engine, when it’s not running the shafts may sit directly on the bearings, but when it is spinning at high speed it is supported by a thin but pressurized layer of oil which prevents any surface-to-surface contact, minimizing wear.
As a practical matter, at least on the high school physics or machine design level we represent solid surface-to-surface friction as a single coefficient for static friction and another one for dynamic friction, or perhaps a curve for dynamic friction. In reality, once you get into the nanomachine/cellular/molecular level, “friction” behavior dominates inertial mechanics and you have to cope with the actual individual mechanisms rather than just pretending that it’s all one big bag. The entire phenomena has to itself the field of study of tribology which is both absolutely fascinating and fantastically boring at the same time, depending on how deeply you delve into it and what your interest is in applied continuum mechanics.
So…it’s complicated; too complicated to sum up in a couple of paragraphs. Anything important usually is. I wish I could offer up some general reference for the layman on the topic, but I don’t have anything in my library that breaks it out at that level.