Managed code is not compiled to machine code but to an intermediate language which is interpreted and executed by some service on a machine and is therefore operating within a (hopefully!) secure framework which handles dangerous things like memory and threads for you. In modern usage this frequently means .NET but does not have to.

Unmanaged code is compiled to machine code and therefore executed by the OS directly. It therefore has the ability to do damaging/powerful things Managed code does not. This is how everything used to work, so typically it's associated with old stuff like .dlls