In the case of floppy disks, the heads generally slide along the surface of the magnetic media. In the case of hard drives, the heads fly above the disk surface at altitude as low as 150 Angstroms (=15nm =6 microinches). The "flight altitudes" are constantly decreasing with improvements in drive technology.
The heads themselves started out similar to the heads in tape recorders -- simple devices made out of a tiny U-shaped piece of highly magnetizable material called ferrite wrapped in a fine wire coil. When writing, the coil is energized, a strong magnetic field forms in the gap of the U, and the recording surface adjacent to the gap is magnetized. When reading, the magnetized material rotates past the heads, the ferrite core concentrates the field, and a current is generated in the coil. The gap where the field is very strong is quite narrow. That gap is roughly equal to the thickness of the magnetic media on the recording surface. The gap determines the minimum size of a recorded area on the disk. Ferrite heads are large, and write fairly large features. They must also be flown fairly far from the surface thus requiring stronger fields and larger heads.
Metal in Gap (MIG) heads are ferrite heads with a small piece of metal in the head gap that concentrates the field. This allows smaller features to be read and written. MIG heads were replaced with thin film heads. Thin film heads are electronically similar to ferrite heads and use the same physics. But they are created using photographic processes and thin films of material that allow fine features to be created. Thin film heads are much smaller than MIG heads and therefore allow smaller recorded features to be used. Thin film heads allowed 3.5 in drives to reach 4GB storage capacities.
The geometry of the head gap is a compromise between what works best for reading and what works best for writing. The next head improvement was to optimize the thin film head for writing and to create a separate head for reading. The read head uses the magnetorestrictive effect which changes the resistance of a material in the presence of magnetism. These MR heads are able to read very small magnetic features reliably, but can not be used to create the strong field used for writing. The term AMR (A=anisotropic) is sometimes used instead of MR. The latest advance in heads is to replace MR/AMR read heads with similar heads using a similar but dramatically improved material technology known as the Giant Magnetostrictive effect. These GMR heads are the current state of the art in 2002.