What are the Tailed Beasts (Bijuu) actually made of? What kind of creatures are these Tailed Beasts? 

I know they are the summation of huge amount of chakra and a body. 

If they are just chakra, can't the chakra itself be the body? 

What do Tailed Beasts need the body for?
Bijuu are living beings, same as humans and animals. They were created by the Sage of Six Paths, using the Creation of All Things technique.

The sage created the vessels with Yin, breathed life into them with Yang, then poured the Ten Tails' chakra into them.

Chakra is merely energy. It cannot take complex physical form (With the exception of the Creation of All Things, which uses chakra to create form).