Investing in Germany

I assume this is about normal, retail investors (not financial professionals or the super-rich, for which other rules apply).
You need a residence visa, registered address and German tax ID to open any kind of investment account with a financial institution in Germany.
Opening an account is usually easy with online banks and brokers, more complicated at offline branches. You might be asked to have your ID papers verified at any post office (the service is called "PostIdent").
You can then put money into your account and, depending on the chosen institution, invest it in a large number of ways, including shares, options, bonds, mutual funds, fixed interest papers, investment-linked insurances, etc.
To avoid costly mistakes, it is best to know what you are doing and to do proper research and comparisons before making a decision. Individualized advice is not available, except at the more expensive offline banks (or from professional weath consultants, for a fee - any person who offers advice for free has a hidden motive - and not your gain - in mind!).
Typically attractive investments in Germany are the many "global champions" - small and medium sized companies that are world market leaders in their (often rather specialized) field, for which Germany is famous.
Property is also attractive, but only for rental returns - prices do not typically increase as fast as in some other markets.
Please keep in mind that all profit you achieve is liable for (high) German income taxes and declaring it correctly will require you to engage a tax consultant (for a fee).

Like Beppi mentioned, to have an account with any financial institution in Germany means being a resident. Once upon a time such things didn’t matter but those times are over. Now many things are scrutinized. One example; I have an account at Commerzbank, one of the countries larger banks. I recently wanted to change a few hundred dollars into euros there as currency exchange offices are always expensive. They mentioned that they ONLY change for clients and the money only gets deposited on an account after a few days to clear. And one has to state where the money came from! Why this should matter for a small amount I don’t know but they have tightened the rules to crack down on money laundering and criminal activity.  Yet one can bring an unlimited amount of money in any currency into the country as long as it is declared if over 10,000 Euros.

Thus at this bank a tourist would not even be able to exchange money and certainly not open an account. As far as I know, a foreign non-resident can still buy property in Germany but I don’t know how complicated this might be with all of the other banking and currency restrictions. In recent years there have been a number of people who asked for advice on this site about buying property in German although they were not even EU citizens or residents. They were just obsessed after having read how much German property values have gone up over the last decade or two. Yes, property has gone up and despite the now high levels seems to still be rising although not as fast as before.

The problem for a non-resident would be that they would not even have a guaranteed right to come to Germany to visit or take care of the property. And finding a good deal on the right property takes some insider and local knowledge. And then to fix up or maintain it one would need workers and /or a property management company, which can likely eat up much of any profit. Not that there isn’t potentially money to be made in this sector but it should be left to people who actually live here or either have a lot of experience or local partners.

In Germany, you could invest very well in real estate a few years ago. The market is very saturated at the moment. Real estate still offers good opportunities to invest.