Selectivity is simply a measure of how hard it is to be accepted at a university. Different universities have different standards for admissions, but the most selective American universities look for applicants with high grades (GPA), a challenging academic curriculum, and (usually) high SAT or ACT test scores, as well as strong recommendations, essays, and extracurricular activities. To put it shortly, the most selective universities want the best students.
A research university is an institution that includes both undergraduate & graduate students and has a focus on both educating undergraduate students & doing cutting edge research with its professors & graduate students. Though exceptions exist, most elite research universities are medium or large in size, private, and are made of up of between 30-70% undergraduate students.
In comparison, liberal arts colleges are institutions that are completely are nearly completely focused on the undergraduate education. They are small institutions usually ranging from just 1000-3000 students that are focused on the undergraduate experience & education. Size and undergraduate focus are the two areas that set research universities & liberal arts colleges apart.
Not all universities have undergraduate degrees in many professional fields. However, that doesn't mean that you can't go into fields like education, health, or agriculture with an Ivy degree! You can study the subject like biology or chemistry for health or English for communication and either learn on the job or get a graduate degree in the more specific graduate field.