University in the USA
First some 'English' ;-)
Americans don't generally refer to 'University' as simply 'University'. "I studied at University" is just not part of American English, just like they would never say "I'm off to hospital"; they would say "I'm off to the hospital". In fact, they rarely refer to a 'university' as a university, unless it is part of the proper name of the school. "I study at the University of Notre Dame". When they are speaking casually, they often just say "school" or "college"--"I'm in school at Notre Dame" or "I'm heading back to college" or the most famous line by John Belushi from the movie Animal House, "Seven years of college down the drain".
Technically, a university is made up of different colleges, with the College of Engineering, the College of Agriculture, the College of Business, the College of Liberal Arts all being part of a "University". Some schools that have only one general degree program, such as a liberal arts degree, are still known as a "college"--St. Johns College in Annapolis, Williams College in Massachusettes, etc. But as far as most Americans are concerned, "college" and "university" are essentially the same thing.
Basic Course of Study
College/university in the US is something most Americans undertake following graduation from high school (at age 18) and pursue for 4-5 years (or 7 in the case of Senator Blutarsky). University education in the US is split between publicly-funded schools (University of Michigan, University of Illinois, California-Berkley, etc) and private schools (Harvard, Yale, Stanford, Notre Dame). There also exists what are known as "Community Colleges" funded by the state and offering Associate degrees (2-year courses).
An American 'Bachelor's degree' (Bachelor of Arts, Bachelor of Science) is generally a 4-year-program. There is approximately one year of "general studies" in which a student will be required to take some basic English, history, math, science and in some schools, religion or theology (if it is a religious university). After that the student focuses on course "in their major" taking courses relevant to their degree (about 2 to 2.5 years worth of courses) and a few 'electives' in which students can get a "minor" in some other field. A majority of students change majors during their 4 years in college after exposure through the general education requirements and electives to other fields of study.
An 'Associates Degree' is a 2-year course of study, most often offered by community colleges and frequently involving work in a trade (auto repair, forensic technician, dental hygienist, etc.).
Community colleges serve several a couple of different roles--they grant associate degrees in some trades for those who don't need a full degree but do require some higher education. They are also a way for some borderline students to get a taste of higher education and decide whether or not they want to attend a full 4-year school. And finally, most importantly, they are a great stepping stone into the higher education system at a much lower cost. Many students will attend a community college to get their "required courses" out of the way, i.e. basic math, English, science, philosophy, etc. Once they graduate with an associates degree, they transfer to a 4-year school (often more prestigious) and go on to study for a full bachelor's degree from that university. The course work they did at the community college transfers over to that school such they save themselves a couple of years of tuition at the big school but still end up with the same diploma.
How much does this cost and what assistance is available?
How much? It changes every year, but for a private school you should budget $50,000 -$60,000 a year in tuition, fees and expenses (food and rent). A public university would have significantly lower tuition but the other expenses would be comparable. Most Americans would budget $15-$25,000 a year for a public university. Tuition is of course only a part of the expense. The basic living expenses also must be taken into consideration. Student housing in New York City (Columbia, NYU) is more than housing in Lincoln Nebraska (University of Nebraska) so these costs will vary significantly.
Financial assistance is difficult to obtain short of citizenship. Some programs exist for green card holders but temporary residents are not eligible for many programs.
A key thing to remember is the concept of "in state"--for a public university, if you can prove residence you are given "in state" tuition rates. Note: residence for "in-state" tuition is an entirely different test than "residence" for immigration purposes. Each state varies.
If you live "in state" you are eligible for a discount at the local public university but if you attempt to attend a public university "out of state" you will pay a much higher tuition fee (comparable to a private university). For example, the taxpayers of Illinois subsidize those Illinois citizens who decide to attend the University of Illinois or Eastern Illinois University or Southern Ill. U. If someone from New York wanted to attended Illinois, they would not receive this local subsidy and have to pay quite a high fee, essentially the same as if they were attending a private school.
This savings can be as much as $20,000-$30,000. Some students attend public universities "in state" and then transfer for the last two years to a more prestigious and expensive private school for graduation.
A university degree is quite valuable in the US. Many employers will simply throw out any applicant who has not received a degree full stop. There are many unemployed college graduates, and even more "underemployed" college graduates so attempting to enter the workforce with simply a high school diploma is quite difficult. Not having a degree in a field of work in which most people do have degrees will be a hinderance, and absent SIGNIFICANT work experience in that field, you will encounter difficulty with hiring and promotions.
One final note: during the junior year of university (third year) many American student "go abroad" and take courses at a university overseas. Most US universities have relationships with schools around the world, including the UK, enabling them to pay US tuition and attend school overseas for a semester or a year.