There basically are only 2 types of printing processes: contact and non-contact.

Contact printing

Uses contact combined with pressure to transfer an image from some sort of an image carrier on to some sort of a substrate.

Examples of “contact” printing: conventional offset lithography, letterpress, flexography, gravure and screen printing. Done using mechanical presses.

Offset lithography is the most common contact printing method.
An image carrier used by offset lithography presses is called a “plate”.
A high-resolution electronic image gets transferred on to the plate using lasers, in a device called “platesetter”. Plate then gets mounted on to a plate cylinder inside the press. When the plate cylinder rotates, it comes in contact with ink and water. The “image” areas on the plate attract ink and the “non-image” areas attract water. An image gets transferred from the plate on to the intermediary carrier called the “blanket”, and then from blanket to paper.
Printing speeds are quite high, and the printed image is always “static” (never changes).
The process is best suited for long runs of magazines, newspapers, books, packaging, etc.

Flexography uses flexible rubber plates as image carriers and used mostly for package printing.

Screenprinting uses a method where the thick paste-like ink gets pushed through the synthetic screen on to a substrate. The ink passe though the sceen only in the areas not blocked by a “stencil”. No plates are used in this case.

Non-contact printing

Transfers an image to the substrate without any contact or pressure.

A few examples: electrophotography (xerography), inkjet, thermal die sublimation. 

These are digital printing processes, and because there are no static image carriers involved, a completely different (variable) image can be printed with every cycle.
This works great when printing addresses, barcodes, other personalized marketing materials. 

Printing speeds are much slower, suitable for shorter / on-demand runs and for variable data printing.

Photocopiers use the electrophotography principle: toner powder (mostly made from finely-ground carbon, polypropylene and various minerals) gets negatively charged and first gets transferred on to the positively-charged image areas on the light-sensitive drum and then get transferred on to the paper.
A high temperature gets applied and melts (“fuses”) the toner into the paper.
The image quality is not as nearly as high as that of the offset lithography presses.

Inkjet printers spray the ink directly on to the substrate from multiple tiny nozzles.
Inkjet image quality can be very high, approaching the offset presses. That is why the printing companies use the high-quality inkjet printers as proofing devices – to closely imitate how the image would look on the actual press sheet. However, because the highly liquid inks can soak the “normal” paper quickly, only specially coated papers can be used.