No actual rules dictate which hats faculty can wear, and you might see a wide variety of hat styles during a ceremony. However, generally the round, eight- or six-sided velvet hats called "tams" are for faculty with doctoral degrees. PhD faculty also have large gold tassels on their tams, called "double tassels."
Flat, four-cornered mortar boards are for lecturers or adjunct faculty with master's degrees. Instructors or adjuncts have smaller tassels that are usually gold or black.
The hoods are long for PhD. faculty. The colors on the silk portion of PhD faculty members' robes indicate where they earned their doctorate; the silk colors are the colors of the school. The velvet stripe around the edges indicates the faculty member's field. For example, white indicates a degree in English, whereas pink indicates a degree in music.
The hoods are shorter for instructors or adjunct faculty with master's degrees. The colors on instructor or lecturer robes indicate where they got their master's degree. Master's degree hoods do not have a velvet section with a degree color. However, some schools have similar colors --- for example, the University of Michigan has maize and blue, but so do many other schools, so you may not be able to tell definitively the school from which any faculty member, whether a holder of a PhD or terminal master's degree, graduated.
The robe for a PhD faculty member is long and typically a matte black fabric. The robe has a wide velvet stripe down the front and smaller stripes running horizontally on the sleeves. These stripes can be black or blue; both colors indicate a PhD. The sleeves may be belled or pointed at the ends.
Robes for instructors or adjuncts are shorter and are typically made from the same shiny black material as the undergraduate robes. They do not have belled or pointed sleeves or stripes.
Faculty can wear whatever they want under their robes as long as it does not show. If pants stick out under the robe, they should be dress pants. The shoes faculty wear are always dress shoes, but no rule exists for color or style. Many female academics take great care in considering the shoes they will wear with their robes since shoes are one of the few items that can accessorize the ensemble.