Does friction depend on the surface?

Yes, friction depends almost entirely upon the surface, or rather, upon both surfaces at the point of their interaction.

Friction is force that resists the relative motion of two bodies that are in direct contact with each other. In more everyday terms, any time two objects are in contact and in motion, such as sliding against each other, anything opposing this motion is friction. Common examples would include the motion of a tire over a road; a less intuitive one is the motion of your feet against the ground. Friction is what makes motion as we experience it in everyday terms possible.

Friction in this context is largely the result of the roughness of the interacting surfaces. Roughness can be thought of as the irregularity of a surface, going all the way to the molecular scale; even a surface that we would describe as smooth in everyday terms is probably irregular at the molecular level. A perfectly smooth object would have, in theory, no friction, and it would be very difficult to interact with except at perpendicular angles.

The more irregular a surface, the more friction we can assume it will generate. Likewise, the greater the irregularities between two surfaces, the more friction they will experience. When it comes down to the actual mechanism of friction, it's largely a matter of those irregular surfaces interacting with each other like the teeth of a zipper - the two objects are essentially bouncing against each other repeatedly, and this motion ends up being tangential to the angle of their relative motion.