Can someone explain European graduate programs in STEM fields? I'm a US undergraduate in math and physics and I'm looking to ultimately do a PhD in either physics or math. It seems that a lot of the research I connect with the most is happening in Europe. I also like the focus on research rather than coursework in European PhD programs. However, I've found most of the process confusing for an outsider like me. 

I would like to know the ins and outs of how graduate programs work over there. There's different lingo there, for example: Taught vs Research MSc. Which is better ultimately for a PhD program? I also get the feeling that Master's and PhD in Europe mean something totally different than what we mean in the US. Are there funded Master's programs in Europe (in math/physics) as there are in the US? How does funding compare?

For reference, I've been looking at schools in UK, Germany, and Switzerland (although not so much Switzerland due to the high cost of living). 

Thanks!

Edit: I'm now aware of how broad my question is. Nevertheless, I found much of your advice helpful in gaining a general knowledge of how to pursue postgraduate degrees abroad and some additional considerations, so thanks! If/when I have more specific questions about this process I will ask them in a different thread.
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