Will stay less than 90 days in Germany, but will work there, what kind of visa do I need? [duplicate]

The principle is surprisingly simple: Since you need a visa and you are staying less than 90 days, you need a regular type C “uniform” Schengen visa. Many people are not aware of it but a Schengen visa is not a “visitor”, “tourism” or “business” visa, it's a short-stay visa and it can in principle cover any purpose, as long as your stay is shorter than 90 days. By contrast, long-stay visas or residence permits like the “employment visa” mentioned in your question are for stays longer than 90 days.

But not every Schengen visa holder is allowed to work, that would be too simple. Depending on the nature of your internship, you will therefore also probably need an authorisation to work. That authorisation does not entail any right to enter or stay in the Schengen area by itself and does not replace a visa (whether short or long-stay). People who don't need a visa for short stays (e.g. US citizens) also need the authorisation but still don't need any visa. Unlike typical work/residence permits, which roll the two things into one, in this case the visa and the “work permit” are two entirely separate things.

In practice, you will need to obtain the authorisation before applying for the visa, as you will have to join a copy of it to your visa application to justify the purpose of your stay (if you apply for the visa directly, without waiting for the authorisation, it will be refused). That process should be initiated by your prospective employer, who should apply for the authorisation on your behalf and then give it to you once it has been approved by the authorities.

Once you have the authorisation, you will apply for a Schengen visa (and not for a long-stay residence/work visa) through the relevant German consulate in the US, using the regular procedure. It's advisable to keep a copy of the authorisation with you to be able to show it at the border if challenged about the purpose of your trip.

Since it can take quite some time to sort out all the formalities, you should contact your employer about this as soon as possible. If they need more info to understand what you need, this document (in German) from the Federal Employment Agency is specifically intended for employers and includes the relevant application form and many details. There is also a form that should be filled by your university (it's the only part in English, at the very end of the document).

See also Is a Pakistani allowed a six-week internship in Germany on a 'normal' Schengen visa?