Are there any other kinds of mildew that affect garden plants besides powdery mildew and downy mildew? If so, what are they?
None - the names Powdery Mildew and Downy Mildew refer specifically to the symptoms present on the plant. Both are caused by various fungi (meaning several different fungi can cause Downy Mildew, and the same for Powdery Mildew) and each variety of fungi is plant specific, meaning it prefers some plants and not others, so the mildew you see on one particular plant may not be caused by the same fungus as that present on an altogether different variety of plant.
Powdery mildew tends to occur in dry conditions, Downy Mildew needs damp and usually cool conditions - both forms occur more frequently on plants which are overcrowded (thus with less airflow around them) with low light levels. Both forms of mildew are spread by airborne spores, though Downy Mildew spores prefer damper air.