The physical composition of dominoes has changed throughout the years. If you read the history of dominoes you will find many articles You can find many articles containing the history of dominoes and the reason they are called bones. The focus of this post is primarily modern day dominoes.
Very Brief History
If you want the history of domino composition you can find a very comprehensive article on Wikipedia https://en.wikipedia.org/wiki/Dominoes. In summary, dominoes are sometimes called bones because they were once made of actual animal bone.
Fast forward to today
Today, dominoes are made of a variety of materials. Because of the maker community being so vast there is no way to make a full list. Someone is bound to make a domino out of something completely unheard of simply because it is not on the list.

Categories of modern mass produced domino materials are Plastics, Metals, Stone and Wood. There are also specialty materials used such as foam for giant yard dominoes pictured here, but those are not the main focus.
Wood

There are two types of wood dominoes. The inexpensive, mass produced for applications where low cost or high quantity are a factor. If you have a classroom using dominoes to make a domino photo mosaic then the dominoes pictured on the left are the way to go. There is also the high end wood domino made by a true craftsman, many times layered in multiple woods and finely finished with layers of lacquer. These dominoes should be considered works of art and will most often have hefty price tags that reflect the time and artistry put into them.
Metals

Metals are the same as wood in the varying quality. There are stamped aluminum dominoes with painted pips that you can find in the promotional product business and hobby shops. However I have seen some fantastic high end aluminum or steel dominoes made and polished or anodized by master craftsmen that can run hundreds of dollars per set.
Stone

I have only seen a few sets of stone dominoes in my life. These were at a rock shop and I still regret not purchasing them at the time. They were made of a marble and were quite heavy even though the individual tiles were quite small.
I was able to find this photo of some but have never been lucky enough to play with such a cool set.
Plastics
When you go shopping for dominoes, the chances are this is the type of domino you are thinking of . Many years ago bakelite was the only option for non bone or wood dominoes. You can still find bakelite in antique stores on occasion.
Similar to wood and metal, you have different types of plastic resulting in different price points of the sets
Inexpensive lightweight dominoes used for domino tricks such as making patterns, stacking and watching them fall. These usually do not even have pips and are not good for playing the game of dominoes. If you are making patterns or toppling dominoes, these may be perfect for you.

Acrylic dominoes. These dominoes are made of 100% acrylic. The sets we carry that are made of acrylic are the marbleized gold and marbleized silver sets of dominoes. Acrylic is used to give them the marble coloring and swirled appearance. You cannot get this appearance with a solid plastic resin.
Click here to personalize your own set of double 6 marbleized dominoes.

Urea dominoes. These are the most common of all the plastics. If you go to a big box store you will find most of the dominoes there are made of this plastic. On our site, all but the marbleized are made of this material. This is a heavy solid feeling domino which engraves beautifully. They come in a wide variety of sizes, colors and sets. What you find in most stores are the 3/8″ thick variety and on our site you will find the 1/2″ thick variety. The different sizes and thicknesses are what make up the weight and price differences you see between different sets.
Click Here to personalize your own set of double 6 dominoes with spinners.