Machines out in the wild can print in anything from plastic filament (hobby grade printers) all the way up to ti64 medical grade titanium (Using a FREAKING LASER to melt layers of the powder together).
Some of the most popular technologies are:
FDM: Fused Deposition Modeling - hobby grade printers, like a hot glue gun attached to a motorized etch-a-sketch with a z axis
SLS: Selective Laser Sintering - using nylon powder (generally) to create models. Laser used to melt layers of the powder together
DMLS: Direct Metal Laser Sintering - metal version of SLS
ExOne: developed by the ExOne company, it uses a binding agent to solidify layers of metal powder which are then baked and infused with a metal like bronse for a decently strong part.
Polyjet: TONS of small jet-based print heads that can make extremely fine detail (down to 16microns) and even print in a few colors and multi-deurometer (spelling?) rubber parts (Stratasys 500 Connex 3)
CJM: Color Jet Modeling. The process that I use in shop. Using HP inkjet heads (c4810a) it lays down layers of colored binder to solidify a gypsum based powder layer by layer. This is the most popular for full color printing.
SLA: Stereolithography - arguably the first 3D printing technology, developed in the early 1980's, uses a laser to cure a vat of photo resin. Extremely detailed, down to 25 microns.
Some printers have hoppers, like mine, that hold the raw material. Consumer grade machines have spools that are either internal or external. All depends on the process.