Plastic tarps, which are commonly called poly tarps because they are made from polyethylene, have become the standard. There are many reasons why plastic has become more popular than canvas, the original tarp material.
Poly material tends to be much lighter in weight than canvas or vinyl. A tarp of the same size can typically fold much smaller as well. These traits make plastic tarps much more portable than other options.
Poly tarps come in a much wider array of options than any other type of tarp.
Some options include:
Material thickness / weight: Poly tarps can range from fairly lightweight and disposable at 2-4 mils thick and under 2 oz per square yard to much heavier, reusable, and durable at 14-16 mils thick and over 8 oz per square yard.
Attachment type and spacing: You generally need some way to secure your tarp, and poly tarps come with a number of different attachment types, including grommets, D-rings, and web loops. Tarps will also vary in their spacing and combination of these.
Color: The first image that comes to mind for many people is a blue tarp, but poly tarps are produced in every imaginable color. Some are even reversible, with a different color on either side.
Reinforcement: There are several features that manufacturers can use to strengthen a poly tarp: rope in the hem, reinforcement patches at grommets and corners, plastic corner reinforcement pieces, double-thick panels, etc.
Special design options: While most poly tarps are straightforward (rectangular and either grommets or D-rings around the edges), there are many special features and designs. Some include multiple rows of grommets or D-rings, non-rectangular shapes, three-dimensional structure achieved with sewing or snaps, a drawstring, and more.
While cotton canvas tarps, polyester canvas tarps, and nylon tarps are usually treated to be water resistant, only plastic tarps and vinyl tarps are 100% waterproof. And while a mesh tarp can provide partial shade, windbreak, or privacy, a solid plastic tarp can perform all of those tasks more effectively.
Plastic tarps are generally the lowest cost option, though cost does vary greatly with quality. This makes poly tarps an easy choice if you're looking for something to have on hand in case of emergency or to use in a situation that might result in a damaged -- and therefore unusable -- tarp.
Other tarp materials certainly have their benefits, but it's clear that there are solid reasons why polyethylene reigns supreme in the marketplace.