In the US there is first a federal (country-wide) rule: there are high-school diplomas, roughly equivalent almost-like-a-high-school-diplomas called a GED or a variety of other names, and then there are accredited higher education institutions providing certificates, 2-year degrees (called an associate's degree), 4-year degrees (called a bachelor's degree), 5-7 yearish degrees (master's degree), and of course the PhD...in addition to a slew of other special professional degrees for high ed.
Where it gets particularly hard to generalize is we have 50 states that determine all the details of the rest, especially the high-schools. I'll try to provide a broad overview. While this is true across states for the most part, the exact mix, nature of funding, and multi-track availability varies tremendously across the country.
Traditional high schools are the most common, which simply teach a variety of subjects and offer some sports, music, and arts programs (where possible), and don't especially differentiate between students who are planning to go to college from students who don't. Some special programming exists tailored to certain professions or college prep, but this is usually minimal. Graduation gives you a plain high school diploma.
Magnet schools exist at the elementary through high school level, and are focused on a specific area of excellence that permeates their whole school design; some schools specialize in science and technology, math, media/broadcasting, dance and/or fine arts, etc. This isn't necessarily about college or professional prep, but there is often a particular bent to some schools and they generally encourage higher education. Graduation gives you a plain high school diploma.
Vocational schools are high schools that are much like magnet schools, only their focus is on certain professional preparation. There are schools that offer programs in all sorts of vocations from culinary arts, electrician, building/construction jobs, auto repair, and many others. Graduation gives you a plain high school diploma.
College preparatory schools (prep schools for short) are, at least in theory, designed with college seeking students in mind. Everything is geared towards being ready for college and getting into a good college. Graduation gives you a plain high school diploma.
There are also some "alternative" institutions that seek to provide a high school education to people who didn't fit in well with other institutions for various reasons (ranging from disability, family situation, pregnancy, legal problems, different learning style, etc), as well as religious institutions. All offer plain high school diplomas, generally.
Now, anyone with a high school diploma and usually even a GED (basically a special program or test that allows one to get a diploma - or something like it - without having to go back to high school or through high school at all) can apply to any higher education institution they like.
Most higher ed sets it's own standards for admittance, though, which is usually based upon grades while in high school (GPA), ACT/SAT test scores, essay, etc. Some places have preferences or relationships with certain high schools (the high schools often being referred to as "feeder schools"), but that's about it.
Now, higher ed is yet another ball of wax, and varies tremendously by state - even more than high schools!
Most states have community, technical, or junior colleges, which generally offer the 2 and 4 year degrees. As these places are often smaller in size, they offer limited and differing selections of degree programs (some may have no fine arts programs, others might have no engineering...etc). Florida calls these smaller places junior colleges, Wisconsin has state technical colleges, Arizona has community colleges...etc. They are similar, but funding and "prestige" varies heavily.
The terms differ, but the general rule is the same: smaller, less degree programs, often focused on certificates and 2-year degrees and very limited 4-year offerings (if any at all). It is not uncommon for the smaller schools to focus on providing a lower-cost, student-centered program with the expectation of transferring to another institution to complete their bachelor's degree.
Then there are the bigger universities, which are sometimes called "4-year institutions", and focus on a larger campus, more degree programs, and are centered on the bachelor's degree. Some offer limited master's programs, and even fewer have select PhD options in specific areas.
We also have "polytechnic" colleges, which vary in size but are often focused on certain aspects of the sciences such as engineering. It is not uncommon for other institutions to omit engineering and related programs entirely, and thus specialized colleges fill in the gaps.
Then there are just big "universities", which offer a wide selection of everything, really. Most states have at least one, and sometimes 2 or more, of these. They often cost more, have more options, are located in big cities, and are just massive.
However, no matter what institution, the same rule generally applies: some are "selective", which means you take tests, and some have open access and anyone who can pay the fee can enroll. The tests are most commonly the ACT/SAT if you are coming from high school, but some schools - especially junior/technical/community colleges - will instead offer their own "placement test" mainly in math and reading to determine if you need to take any remedial classes first.
Selective schools limit enrollment, and usually use a multiple factor process: personal essay, maybe reference letters from educators, transcript of grades from wherever you went to high school, ACT/SAT scores, etc. Each school sets their own rules, and this ranges from your MIT schools who are proud of their tiny admittance rates, to most schools who just don't want to admit students they think will not thrive in their programs.
Students who can't get into selective schools but want to are often encouraged to attend the junior/technical/community colleges for a year or two, and then re-apply. When applying then the high school grades and tests are at times ignored completely or at least heavily discounted, and strong preference is given to those with good grades at whatever college they have been attending. This process is very common with mid-to-large state-funded schools.
So, in summary: high school diploma is just a diploma in the US, higher education institutions set their own individual admittance standards (ranging from strict to none, varying by place and state), most selective institutions require ACT/SAT scores and high school transcripts at a minimum regardless of where you went to high school or what you did there.
I hope this provides you with more than you actually wanted to know!