Offset printing method
The first ever press was a Letterpress. It basically uses stamps to grab ink and place it on the paper (or other material). Think of a typewriter, but doing whole pages in one press. Of course, this took long to do as each page was setup before by hand and manually placing these letter stamps in place. As time went on, full page stamps were created instead to make the process easier (though still inefficient compared to other methods). Today however, the Letterpress is not used much as it is not an efficient and far too expensive method of printing.
This method of printing is the most common used today. It is also one of the oldest. It works on the principle that water and oil (ink) don't mix. Using metal or polyester sheets (called plates), image and non-image areas are burned onto the plate using light to expose the image areas. this plate is attached onto a cylinder that as it goes around on the press, picks up water onto the non image areas. since water and oil don't mix, when the plate comes into contact with the ink, it only sticks to where the water isn't, our image area. The plate then comes into contact with a rubber sheet (called a blanket) and it transfers the image. The blanket them rotates around and presses to image into the paper. This is where the offset term comes from. While other methods can be done with offset theories, the Offset Lithography is so common that when someone refers to offset printing, this is what they mean.
This is traditionally used to print labels. If you look at a bottle of pop, the plastic or cellophane label on it was likely done by flexography. It is the packaging industry who primarily uses flexography. The idea behind flexography is similar to a Letterpress where it uses a stamp, but this one is created with rubber etched with tiny grooves that pick up ink. The rubber stamp (plate) wrapped around a cylinder which rotates and picks up ink from a reservoir then presses it into the printing material. This is often done on plastics, tissues, labels, stickers and cardboard.
This is a method usually used in printing long runs of magazines. Much like flexography, gravure printing has a cylinder that picks up ink in tiny etched grooves and places it on the paper. The difference is, gravure doesn't use a plate. Its grooves are actually etched into the cylinder. This allows it to last much longer and can be used for more impressions (contacts with the paper/material) before it wears out.
This is still a common method of printing. It is often used on all the odd materials. Solid letters on plastics, T-shirts and clothing materials, a lot of signs and others use screen printing. The idea behind screen prints is basically a screened material such as silk or nylon is stretched across a frame and fastened into place. A stencil, cut my hand or made electronically, is placed over that screen to block out non printing areas. Ink (often rubber based) is placed inside the frame and scrapped across the stencil with a rubber squeegee. The ink goes through the screen and onto the material.
There are several way to do digital printing. Many methods try to reproduce the effects of the previously described styles. There are inkjet, laser and toner, and magnetic digital printers. In inkjet, the ink cartridge holds liquid ink that is released in tiny sprays onto the paper . It makes several dots, that when viewed without a magnifying glass creates the illusion of your image. Laser and toner method uses a laser to charge the paper in certain areas which will attract toner of cymk colours to it. It then goes through a fuser which melts the toner into the paper. Magnetic works in much the same way but instead of electrical charges, it uses magnetic ones. It also passes through a fuser to melt the toner on.
This is all just touching the tip of the iceberg for each of these methods. This is just a rough summary of what these printing methods are all about. If you are interested in learning more, the internet has all the information you could want about this. Printing is considered the greatest invention of the second millennium and is a major reason why we are where we are today.