Materials that can be laser engraved and cut:
Cardboard
Ceramics, including Dishes, Tile, etc.
Glass
Paper & Paperboard
Rubber
Genuine Leather
Stone, including Marble, Granite, River Rock, etc.
Textiles, including Cotton, Suede, Felt, Hemp, etc.
Wood, including Cork, MDF, Plywood, Balsa, Birch, Cherry, Oak, Poplar, etc.
Plastics including:
Acrylonitrile Butadiene Styrene (ABS)
Nylon (Polyamide, PA, etc.)
Polyethylene (PE)
High-Density Polyethylene (HDPE, PEHD, etc.)
Biaxially-Oriented Polyethylene Terephthalate (BoPET, Mylar, Polyester, etc.)
Polyethylene Terephthalate Glycol (PETG, PET-G, etc.)
Polyimide (PI, Kapton, etc.)
Polymethyl Methacrylate (PMMA, Acrylic, Plexiglass, Lucite, etc.)
Polyoxymethylene (POM, Acetal, Delrin, etc.)
Polypropylene (PP, etc.)
Styrene
Genuine leather can be safely laser cut or engraved, but ensure that if it has been tanned then it is “veg-tanned” otherwise it will have been treated with chromium salts that will release dangerous particles when laser engraved.

Materials that cannot be laser cut or engraved:
Artificial Leather containing Hexavalent Chromium (Cr[VI])
Astatine
Beryllium Oxide
Bromine
Chlorine, including Polyvinyl Butyrale (PVB) and Polyvinyl Chloride (PVC, Vinyl, Cintra, etc.)
Fluorine, including Polytetrafluoroethylenes (Teflon, PTFE, etc.)
Iodine
Phenolic Resins, including various forms of Epoxy
Polycarbonate (PC, Lexan, etc.)
Of particular note is anything made from PVC or Vinyl. Not only are the fumes toxic, but they will also condense inside the laser cabinet and cause rapid and premature corrosion of metal surfaces, bearings, etc.
You can mark metals with a CO₂ laser with the help of a special coating agent. However, you cannot cut or engrave metals with a standard CO2 Laser due to their conductivity and reflectivity.