What is ConEmu?
It aims to be handy, comprehensive, fast and reliable terminal window where you may host any console application developed either for WinAPI (cmd, powershell, far) or Unix PTY (cygwin, msys, wsl bash).
What is Cygwin?
It is a POSIX-compatible environment that runs natively on Microsoft Windows. Its goal is to allow programs of Unix-like systems to be recompiled and run natively on Windows with minimal source code modifications by providing them with the same underlying POSIX API they would expect in those systems.