Can someone explain European graduate programs in STEM fields?
I can give you some insight in Germany/Austria (phd in physics in Austria), not so much on the UK. Many countries in Europe now use the Bologna system
https://en.wikipedia.org/wiki/Bologna_Process#Qualifications_framework

It is normally a three step process for a phd: Bachelor programs is more or less equivalent to US undergrad, takes 3 years normally (all numbers with some variation depending on country and field) and ends with a bachelor degree BSc. Step two is a Master program, duration normally two years and mostly course orientated. Normally ends with a master's thesis, which involves independent research in a group under the supervision of a professor. This is a requirement for entering stage three, the phd. In general there is no funding for the master's years (maybe some for the half year during your thesis or if you TA), you are considered a student. Tuition fees depend on country/university and can be waived with stipends if you qualify (no fees in Austria though)

PhD is almost exclusively research in a specific research group and normally fully funded. Minimum duration is three years.

You see, that what you call graduate studies/phd in the US is separated into Masters and PhD in Europe, you obtain a Masters degree in the middle.

This also means that the application process is very different depending on stage. Let's say you want to start your masters in STEM in Europe. Setting visa considerations aside, for some universities you can just sign up for a Masters in Physics for instance, no entry exams. Some might require exams and charge you tuition fees.

One your done with you Masters, you can look for phd positions at either the same University or wherever you like. The availability of positions is tied to funding, which is normally under control of the leading professor of the group. So you apply to her/him and if they like you they can hire you. In most cases you don't have to get accepted by the university, you just enroll. A professor has to accept you. (Very different to the US). Often, you end up doing your phd within the group you wrote your master's thesis in.

On top of that there are graduate schools with extra funding where you apply to the program and the will assign you a supervisor.

So overall: Bacc-Master-Phd; 3-2-3 years