When applying for a Schengen visa, which box do I need to check if my father is the sponsor?

The answer depends on whether you are visiting your father in Germany or if he is simply paying for your trip.

There are three types of sponsor (host, company or organization). Should I check this box?

If you are applying for the visa so you can visit your father in Germany, then you should check this box.

Or, should I check the 'other' box, and then specify that my father will be sponsoring me?

If you are not applying for this visa to visit your father, you should check this box and specify that your father will be sponsoring you. This is true if you will be traveling to the Schengen area with your father or without him.

Do I need to attach a letter from my father? If so, is there any specific template for that letter, or just a simple word document will do?

Requirements for supporting documents are covered in Article 14 of the Schengen Visa Code. On the subject of sponsorship, the article says

Member States may require applicants to present a proof of sponsorship and/or private accommodation by completing a form drawn up by each Member State.

Therefore, you must find out from Germany what their requirements are. When I looked on their web site, I couldn't find anything other than a promise that I would receive additional important information after booking an appointment. Whether that information includes guidelines for supporting documents is not disclosed.

You can assume that you will need to include copies of your father's bank statements, since they will examine his finances to see whether he can afford to pay for your trip. A letter from your father indicating his willingness to pay for your trip will also help, and it should probably be notarized. There does not appear to be a specific form for this purpose.