When using the word Principal to describe the head or director of a school, under what situations should one capitalize it, and when should it be lower-cased?
Principal should be capitalized when used as a title preceding the name of the person but uncapitalized if used as a description following the name. For example,
Let us welcome Principal Bob.
Let us welcome Bob, the principal of the school.
See for example Blue Book of Grammar and Punctuation.
Tips on the capitalization of the word principal:
you should capitalize the word principal when it goes before a proper noun, such as "Principal Skinner", or "Mr. Skinner, Principal of Springfield Elementary School".
you should not capitalize the word principal when the word stands alone without a person's name like "we're sending you to the principal".