they all don't connect to the manifold most do though.. like he said some can be connected closer to the back of the car anywhere on the exhaust, they also muffle exhaust sound so need for a muffler is lessened. they look like snails and suck in air on one end and exhaust gases power it on the other end. they also have things called downpipes(from turbo to rest of exhuast when mounted on the header) and uppipes(from turbo to intake) that naturally apsirated cars don't have.
What kind of turbocharger doesn't connect to exhaust manifold?
Size of the exhaust piping depends on your application and power goal.
For 2.0L 4 cylinder engine, 2.5" should be enough for ~300 hp at the crank.
Any higher than that 3" is recommended.
Smaller piping give better low end while larger will free top end power.