Blatant literally means "noisy." By extension it refers to something done obtrusively and noticeably. The adverb form is blatantly. The noun form, blatancy, is rarely used.
Flagrant means "openly scandalous or notorious." The adverb form is flagrantly. The noun form, flagrancy, is not as common.
The Latin legal term flagrante delicto means "while the crime is or was committed," or as we say in English, "in the very act."