The internal wastegate; is a built-in bypass and opens a gate inside the turbo to let exhaust bypass the exhaust turbine wheel. It allows excess exhaust pressure to dump into the downstream exhaust. It keeps the turbo from building to much boost. Most work off of pressure, some are computer controlled.   Most stock engines the internal wastegate works just fine. When you put a bigger, compound turbos or add more horsepower on then you need to start to think about a external wastegate. The advantages of a internal wastegate include simpler and more compact  installation, with no external wastegate piping. They have limited ability to bleed off boost pressure due to the relatively small diameter of the internal bypass valve.
External wastegate dose the same thing but is different, it is mounted out side the turbo housing before the turbo. You then need pipe to bypass the turbo. Most of the time it go's back in to the exhaust. Race cars and people that don't mind load will dump it out a pipe it is called a"divorced" wastegate.  This is done to prevent turbulence to the exhaust flow and reduce total back pressure in the exhaust system. Back pressure is what causes high ETG's and can harm your engine. Wastegates can be set up a lot of different ways but achieve the same thing
A internal and external can be used together but most of the time if a external wastegate gets installed and the turbo has a internal wastegate the internal will get blocked off some how or welded closed. There is more control with a external wastegate. They come in a lot of different sizes and most have springs that can be changed to open at different pressures. It is important to get the right size, it is best to talk to a wastegate suppler and tell them what you have and what you want to do. You can put two wastegates on one exhaust housing, they can open at different times or at the same time for more control.
I have compound turbos on my Dodge cummins and I have a external wastegate that bypasses my first turbo and then dumps back before the second turbo. The reason is the first turbo has a small housing for quick spooling,  the second (or primary)  has a big housing for big boost. If I let everything go in the first turbo the housing is to small and it would build so much back pressure it would destroy the engine. This is one case where the wastegate is used to re-leave back pressure.

External and external wastegate explained

What wastegate do I need

There are two types of wastegates one is internal and the other is external, they are as they sound

Left is internal right is external
See the video on how to install a wastegate
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