Well, the only quick answer to this is that plywood can be any wood. Rather than being made from one specific type of wood, plywood is rather a method of construction and many different species of wood are turned into plywood.
So what is plywood then?
Plywood is created from thin sheets of wood glued together under high pressure. The sheets have their grains swapped 90 degrees on each sheet. What this means is that the finished sheet of plywood has the grain running in both directions. This massively helps to increase the strength of plywood.
Large logs are effectively peeled into thin strips, these strips are then glued together to create a plywood sheet.
What woods are commonly used?
Hardwood plywood is commonly made from oak, mahogany and beech. Being constructed from hardwood it is incredibly strong and resistant to wear and tear. Like everything hardwood though it is prohibitively expensive.
There is one simple reason for the high price for anything hardwood. Hardwood trees take a long, long time to reach maturity. So production of hardwood is a slow process, so it is always going to be expensive.
Softwood plywood is usually made from cedar, Douglas fir, spruce and pine. This type of plywood is usually used in construction. It is rarely used in furniture and cabinet making, this is due to it being easily dented due to being constructed from softwood.
Many of the cheaper plywoods you find in the big DIY stores are made from tropical hardwoods. This wood is usually cheap but strong, making it perfect for plywood. Often Eucalyptus can be used for the core with tropical wood used for the exterior veneer. These are often red/brown in appearance.
The problem with tropical plywood is simply deforestation. Many of the countries where the wood is harvested are poorer countries with little or no managed forestry. Huge areas of forest have been cleared to create plywood that is now dols around the world.
Birch plywood is made from birch, duh. It is popular due to its attractive light finish and high strength. You can find out more about birch plywood here. It is grown in European managed forests and is considered to be a renewable resource due to its speedy growth and proper forest management.
This type of plywood is intended for applications where the top layer will be on show. The core is made from cheaper woods and is then finished with a final veneer of luxury wood such as oak, cherry, mahogany etc. With this type of plywood, you get a beautifully finished wood on the surface without the cost of solid wood.
The problem with these types of plywood often comes down to the thickness of the veneer. Thin veneers are commonly used, meaning a deep scuff or gouge will go right through and show the layers below. Another problem with thin veneers is that you can not sand them down and apply a different finish as you run the risk of sanding straight through the veneer.