You have a wide variety of materials to choose from, each with their own uses and benefits.
Plastic: These are made to be as washable and water-resistant as possible. However, these typically aren't the cheap plastic placemats you'd find at a children's restaurant. These can often be made to look classy, like they have a woven texture, for example. Choose plastic placemats for ease in washing and style.
Cork: This is a fairly durable placemat material that is resistant to stains and wrinkles. However, it tends to be hand washed only. Cork is also good for not sliding around a table too much while protecting the table from scratches. Cork mats also tend to be heat-resistant.
Polyester: This durable material is easy to wash. Many are water-resistant. Polyester can also be made to have different textures to it, like a classy tweed look.
Mixed materials: Some styles come with fronts made from materials like polypropylene with a foam backing. Typically, you choose mixed materials because those materials impart a sense of style. Though the materials may impart their own use, like foam makes it harder for the mat to slide around. Like any good placemat, these are often made to be stain-resistant and water-resistant.
Cotton: You might also find a placemat made of cotton because it gives a soft texture and is machine washable. However, cotton may be more prone to staining.
Jute/rattan: Choose this placemat for a classy woven texture, which works well in rustic and classic spaces. However, these are often dry clean only.
Recycled materials: If you're eco-conscious, try a green product that makes use of recycled materials. For instance, some mats are made from repurposed wood or leather. These are often treated to be heat, stain and water-resistant.