How is it that some college athletes can be 25, 26 years old while most are the age that you would expect of people recruited out of high school? (18-22)
Students are permitted to participate in college athletics at any age as long as they follow the rules.
The most relevant rule for NCAA Division I athletes is the Five-Year Rule:
Title: 12.8.1 - Five-Year Rule.
A student-athlete shall complete his or her seasons of participation within five calendar years from the beginning of the semester or quarter in which the student-athlete first registered for a minimum full-time program of studies in a collegiate institution, with time spent in the armed services, on official religious missions or with recognized foreign aid services of the U.S. government being excepted. For international students, service in the armed forces or on an official religious mission of the student’s home country is considered equivalent to such service in the United States. (Revised: 4/2/10, 7/31/14)
Most students follow the traditional path of high school followed by college, and so the Five-Year Rule means that their eligibility ends around age 23.
BYU frequently has athletes who complete a two-year mission in-between high school and college, and so often their roster contains older players.
Brandon Weeden, born October 14 1983, didn't register for college until 2007 (as identified in mastermind_ed's comment), so his clock didn't start until he was age 23.