The compressor section is identical in function to any centrifugal supercharger, the only difference is that the turbine section of the turbo drives it. One thing to know is that turbocharger compressor sections are (generally) significantly smaller than their supercharger cousins. This all has to do with efficiency and the chosen method of powering the compressor, so just know it's the reason why you see turbochargers spinning such high RPM when compared to their centrifugal supercharger cousins. It's all about necessity.
This section bears a strong resemblance to the compressor section for a reason; it basically functions the same but backwards. The two main parts are the turbine housing and turbine wheel, and if this is an internally wastegated turbo, the wastegate also resides here (there will be more on that later). As exhaust gasses quickly move out of the cylinder and into the exhaust manifold, they are routed into the turbine housing's scroll. If you understood the flow of air through the centrifugal compressor design discussed earlier, here it's just the opposite occurring. As the hot and rapidly moving gasses attempt to find an airflow path through the turbine housing (with the ever decreasing scroll area), they come in contact with the turbine wheel on their way to the center outlet of the housing. As they rush through this airflow path and into the exhaust downpipe, they spin the turbine wheel, imparting a portion of their kinetic energy to the turbocharger. Especially notice that with this design comes variable RPM, the turbocharger itself is not physically strapped to any rotating part of the engine. This makes many different turbo shaft speeds possible at a single engine RPM, which is where the system's basic performance characteristics and tunability are born.
Although I say "basic" here, know that this is pretty much an oxymoron when dealing with turbos. There is nothing basic about a turbo system, as many different things concerning engine operation need to be addressed. The basic turbo system should come with a bunch of different things, and few systems effectively address all these unless your car was originally equipped with the system. Here they are, in no particular order (with the little things like vacuum line omitted), and notice
2- exhaust manifold for turbo
4- blow-off valve (aka bypass valve)
5- lines for oil supply and return
6- intercooler (optional)