I’m a permanent resident (Green Card holder) in the USA and I’m flying to Jordan through Frankfurt. If I would like to enter Germany for a short visit, maximum 7-8 days, can I apply for a transit visa that allows me to enter Germany or do I have to apply for a Schengen visa?
As already mentioned in the comments, what you plan (7-8 days) is not a transit. Holding a green card itself doesn't allow you to enter the Schengen area without a visa, so it's up to your nationality. In this document you can find a list of all countries whose nationals are required to be in possession of a visa. Europe.eu also shows an interactive map where you can see if you need a visa or not.