My understanding is that, in broad strokes...
Aurora acts like a proxy or mirror that doesn't require you to sign in to get Google Play Store apps. It doesn't provide any other software besides what you specifically download from it, and it doesn't include any telemetry/tracking like normal Google Play Store would.
microG is a reimplementation of Google Play services (the suite of proprietary background services that Google runs on normal Android phones). MicroG doesn't have the bloat and tracking and other closed source functionality, but rather acts as a stand-in that other apps can talk to (when they'd normally be talking to Google Play services). This has to be installed and configured and I would refer to the microG github or other documentation.
GrapheneOS has its own sandboxed Google Play Services which is basically unmodified Google Play Services, crammed into its own sandbox with no special permissions, and a compatibility layer that retains some functionality while keeping it from being able to access app data with high level permissions like it would normally do on a vanilla Android phone.
