Colleges & universities in the US are not standardized. Though many colleges have similar styles and rules and customs, every college gets to do whatever it wants. Just for an easy example, some schools do not have grades or report cards at all. Some leave certain classes off of report cards. Some put every piece of school related information on their report cards. So it's hard to say what "requirement designation" or "flexible core" are unless we know which college the card is from, because those terms could be different at every school. But if I had to guess - a lot of schools have what they call "core requirements" (or something similar). Those are classes that everyone has to take, no matter what their major is. Music students, accounting students, literature students, engineering students, biology students, etc. will all need to take one year of art, one year of math, one year of a foreign language, one year of science, one year of ______, basically the list of things that the college thinks that everyone should learn no matter what they specialize in. I would guess that a "requirements designation" for a class is a note or code that means "this geology class meets the science requirement." I would infer that "flexible core" means that the college allows students to make some choices about their required classes.

"Requirement designations indicate courses that fulfill any combination of liberal studies, multicultural, oral and/or computer competency requirements. NOTE: Requirement designation codes beginning with the letter H indicate an honors requirement designation." http://www.sc.my.fsu.edu/students/how/search-classes-requirement-designation