MME, Windows DirectSound or Wasapi

MME is often the default selection since it is supported by most Windows OSes (MME was released in 1991). Between DirectSound and WASAPI there is not a big difference, as DirectSound is basically just a DirectX-related Interface to the Windows Audio Session API (WASAPI) underneath. WASAPI features the lowest latency of all (by design) and therefore should be preferred for recording (especially when it comes to multi-track).

According to Audacity:

"MME: This is the Audacity default and the most compatible with all audio devices.

Windows DirectSound: This is more recent than MME with potentially less latency.

Windows WASAPI: This host is the most recent Windows interface, that Audacity supports, between applications (such as Audacity) and the soundcard driver. WASAPI was first officially released in 2007 in Windows Vista. WASAPI is particularly useful for "loopback" devices for recording computer playback. 24-bit recording devices are supported. Playback is emulated using this host. As a result, the playback slider in Mixer Toolbar will only scale the system playback slider's current level up or down rather than directly manipulating that system slider."

Everything between the quotes came directly from Audacity. Seems they had a lot to say about WASAPI.

MME (Multimedia Events, aka WinMM [Windows Multimedia APIs]) should be an old interface that exists since as early as Windows 95 (or even earlier).

WASAPI comes with the new audio stack on Windows Vista and higher. As mentioned by Jan in his answer, the WASAPI provides the minimal latency. On my computer, sending audio data to DirectSound seem to have a built-in buffer, so that playback from Audacity is more stable on DirectSound than on WASAPI. But for recording, Audacity using WASAPI captures the audio perfectly.