Both inkjet and laser printers can print on specialty papers, but individual printers may be limited in the media they can print on, or the thickness of the media you can feed through the printer. Specialty papers include all types of photo paper, card stock, transfer or iron-on paper, acetate, parchment, paper with a self-stick back, which includes labels, magnet backed paper, and even fabric specially prepared to go through a printer.
You can find special media in many places, from craft stores, discount department stores, and office supply stores to art supply shops, but you will probably not find all types of media in a single location. Online, there are also many suppliers.
Printers which have a manual front feed that allows paper to move through the printer in a straight path rather than curving around through the printer are the easiest to use with specialty media.
Indeed, some printers have the feature specifically to allow specialty paper use. This includes some photo printers that also have special inks, however many of those printers are at the very high end of most people's printer budgets. Epson and Canon both make printers of this type, and on the right papers they produce photographs of professional or Fine Art quality.