Clay blocks are found in abundance inside lush caves, less commonly in shallow surface water, or can be created from mud with the help of pointed dripstone. When smelted, it turns into terracotta. You can also turn it into clay balls when broken, which can be smelted into bricks.
A block of clay (via screenshot)
Clay can be found commonly underwater in swamps, beaches, rivers, and lakes. They are also generated in lush caves as blobs and are most commonly found here. When broken with shovels, you will receive four clay balls, and this amount can be increased by using a shovel with the fortune enchantment on it. If you want to get the whole block you will need to use a tool with the silk touch enchantment on it. Another way to make clay is by placing mud on top of a block with a dripstone below it. Over time, the mud will turn into clay.
Clay blocks found underwater (via screenshot)
The primary use of clay is to either make terracotta or store clay balls. Clay balls are smelted in furnaces to make bricks. Terracotta is used as a decorative block whereas bricks can be used for both decorating and construction.
Clay balls smelted into bricks (via screenshot)
Clay is not widely used as it offers only terracotta and bricks which are not used for a lot of stuff. People who like to decorate their buildings and make them look more earthy use clay for bricks and terracotta. Other than that, you can also grow cacti on them but they serve as an alternative to sand which is easier to come by.
A. The best place to find clay are beaches, shallow rivers, and lush caves.
A. Clay is supposed to be common but it has very specific requirements to generate. It must be in sand and close to water.
A. Clay blocks are smelted to make a terracotta block, and clay balls are smelted to form bricks.