A Beauty Pageant is a contest to see who is the most attractive which can consist of talent competitions and more. Normally, usually just used for fan service in manga/anime.
A beauty pageant is most known for being a highly glorified popularity contest based mostly on appearances or other superficial maters. The winner is either decided through popular vote of the masses or done by a panel of judges.
The most common practices in any beauty pageant are performed in a minimum of three contests. Those are normally Night Gown, Swimsuit show, and Talent portion. During one of the stages, it's at time customary for contestants to answer questions at random or from judges.
A Beauty Pagent in Japanese is often referred to by a few different names. More often called a Beauty Contest(ビューティーコンテスト). Also called a Miss Con(ミスコン) or Miss Contest(ミスコンテスト)