The most common type of print people get is a C-print, also known as a Digital C-print or sometimes called a Lambda print or LightJet print if the print is made using a Durst Lambda print making machine or an Océ LightJet print making machine.
C-prints are made with dye instead of ink.
Some C-prints, mostly used for advertising and art, are made on clear plastic that is then lit from behind.
C-print papers are coated with light-sensitive material - a silver halide emulsion - that is then chemically developed to activate three main dye layers—cyan, magenta, and yellow. It's a reaction between two chemicals that create the color dyes that make up a photographic image.
The emulsion responds to both exposure and development. A blue-light-sensitive layer forms yellow dye, a green-light-sensitive layer forms magenta dye, and a red-light-sensitive layer forms cyan dye. A series of processing steps follow, which remove the remaining silver and silver compounds, leaving a color image composed of dyes in three layers.
The 2 major brands of C-print paper are Kodak Endura and Fujifilm Crystal Archive, which is how you can tell if the lab making your prints is making you a C-print.
Inkjet printing is a whole different ball game.
Both inks and dyes are used and can be aqueous or solvent based.
Some inks/dyes dry, some have to be heated, and some are cured with UV light.
Home inkjet printers generally have just 4 colors of ink/dye - Cyan, Magenta, Yellow, and Black.
High end inkjet printers have intermediate ink/dye colors up to a total of 12 to 14 colors.
As Tim mentions there is a very broad range of papers we can choose from and we can also get prints made on other media - like canvas, metal, wood and acrylic.
The texture of a inkjet paper (and to some extent how absorbent the paper is) is described as how much 'tooth' the paper has. The more tooth a paper has the rougher the texture of the paper.
One last note.
If you are going to frame a print (have a print framed) with glazing (glass or acrylic) in front of the print you want to make sure the print does not touch the glazing. As indoor humidity fluctuates a print in contact with glazing can become stuck to the glazing. A mat between the print and the glazing, or glazing spacers keeps the print away form the glazing.