A “likely letter” is a letter from the college admissions office stating that you are likely to be admitted as long as nothing changes before the official admissions deadline. They are not formal offers of admissions–just notices that students are very “likely” to be admitted. Students receive likely letters a month or two before the regular admissions decision deadline.
Likely letters are common for athletic recruits to the Ivy League. Since Ivy League schools don’t award athletic scholarships, they use likely letters as a means to indicate to the athlete that they will be accepted at the school. The Ivy Manual allows Likely Letters to be sent out after October 1st and they must come from the admissions office (pg 146).
(List at end updated for 2023.) There are 338 women’s D1 soccer schools in 32 conferences according to the 
The term “Public Ivy” was coined by Richard Moll in Public Ivies: A Guide to America’s Best Public Undergraduate Colleges and Universities which was published in 1985. Moll identified eight public institutions with Ivy League characteristics. He argued that these characteristics, which included both academic and non-academic elements, provided an Ivy League experience at a public-school price. Since then, others have added to the list or often just use the term to describe a public institution with high academic standards.
(Updated for 2023) If you want to get any financial aid, you need to submit college financial aid applications such as the FAFSA and PROFILE. This is obviously a critical step: no application, no aid. But if you want to pay less for college, it certainly isn’t the first step.