The eligibility criterion for appearing in the UPSC Civil Services Examination includes being a citizen of India, a minimum age of 21 years, and holding a bachelor's degree from a recognized university. There are specific limitations on the number of attempts based on category, with general category candidates allowed six attempts, while OBC candidates have nine and SC/ST candidates have unlimited attempts until the age limit. Meeting these criteria ensures that candidates possess the academic foundation and maturity required for civil services.