Offset--used for printing nearly everything, offset uses a flat plate with water-accepting and water-rejecting areas. The plate is dampened then rolled across an inked roller. The ink sticks to the water-rejecting areas. The plate is then rolled across a "blanket" cylinder, which is pressed against the sheet.
Flexography--narrow-web flexo is for packaging, wide-web for newspapers. A plate that looks a LOT like a rubber stamp is inked with an anilox roller, then pressed against the sheet.
Letterpress--used largely for packaging, it's a little like flexo except the plates are made from metal.
Rotogravure--used for million-impression runs of packaging and publications, a polished copper cylinder is engraved, inked and pressed against the sheet.
Wide-format inkjet--used for billboards and vehicle wraps, these are like the inkjet printer you may have attached to your computer, but much larger.
Narrow-format inkjet--used for addressing.
Digital presses--these are getting very popular. These are laser printers that run hundreds of pages per minute. Some print in color.
Intaglio, or Engraving--elegant invitations and banknotes are printed this way. They engrave the information on a flat copper plate then press the stock onto the inked plate.