First comes the application process. When you find a uni you're interested in studying at you'll have to send in the required paperwork (for me it was high school/college transcripts and diploma along with the application form) to an organization called uni-assist. All documents have to be authenticated (called an apostille), which for me entailed taking my notarized school records to the sec of state building in my state to have them stamped with an official seal certifying they are indeed legit.
After your documents are sent you wait for uni assist to classify them and say whether or not you have the minimum requirements to study at the uni and if you meet the requirements they forward your documents to the uni for admission evaluation.
If you're admitted now comes the fun part of trying to find a place to live and get everything sorted out for actually moving over there. As an American it was nice to be able to go to DE on my passport without needing a visa (can do so for 90 days), this simplified things a lot as some people from other countries might be required to obtain a visa beforehand which is just another hurdle to jump over. So as an American I was able to come here and get most other things sorted out first then go apply for my residence permit.
The residence permit requires several things:
1. You be enrolled in a German uni
2. You have health insurance
3. You have a minimum of x euros in what is called a Sperrkonto (blocked bank account, only y euros can be withdrawn per month, I think you have to have a little under 9k euros in there to show you can sustain yourself which will get you a 2 years residence permit)
4. You have a place to live registered with the city (Meldebestätigung I think it's called, this document is extremely important for opening a bank account, getting health insurance, etc.)
5. The correct forms filled out and maybe something else I'm forgetting
So the order I recommend doing things in is:
1. Find a temporary place to live first, check wg-gesucht.de so that you can get the Meldebestätigung
2. Go open a bank account (requires Meldebestätigung)
3. Get everything sorted with the uni, registration etc. and might even need the registration certificate to show for getting student health insurance
4. Get health insurance (requires Meldebestätigung and a bank account)
5. Finally after all that is done go to the Auslanderbehörde and get a residence permit
The uni should provide assistance for getting all of this done and I highly suggest taking advantage of that.
My first uni experience was in the US at a large state university. In my experience the German system is fairly different, but in a good way. My degree is only 3 years as opposed to 4 like in the US because I don't have to take any core classes like I did in the US (history, art, other topics that I found to be a tuition money grab). It's just math, physics, and some electives which are math/physics related or a German class.
The unis here are state funded and you only have to pay some administrative fees which for me is roughly 200 euros/semester - much better than the $4-5k that I paid in the US for tuition and such. Not having as much funding I see it as somewhat of a good thing since the uni can't waste money on useless stuff like a football team or other crap that doesn't belong at an academic institute (just my opinion). It feels more serious, but that could just because I'm older and I take it more seriously. Overall though I really like it and it has been a great experience.