A quick description of each:

Standard - Uses only Lightroom sliders: Exposure, Contrast, Temperature

Advanced - Uses all of ‘Standard’ sliders in addition: Clarity, Cropping, Highlight detail, Lens Profile Correction, Noise Reduction, Shadow detail, Sharpening, Straightening and Vibrancy

Custom -  Uses all of ‘Standard’ and ‘Advanced’ sliders in addition: Finish, Grain and Vignette 

You can also see all the differences within your 'Preferences' section.

****Please note Color Correction is only processed within Lightroom.