Which category of visa do a spouse and child require when accompaning an Indian going to Germany on business?

It is almost the same like in "Taking wife on visitor visa while I am on business visa (UK)", but as we are speaking of the Schengen area which is different from the UK, it is not a duplicate.

Your spouse and child need Schengen short-stay visas. Under the following link you can find necessary information about acquiring one, it is the official site for Germany:

http://www.auswaertiges-amt.de/EN/EinreiseUndAufenthalt/Visabestimmungen_node.html#doc480844bodyText7

Short summation: You need to print out the application and fill it out. Important and necessary: Photo, passport with 2 blank pages, return ticket, travel visa insurance policy, proof of accommodation, bank statement.

Full document of requirements for tourist visa here as PDF

Please apply as soon as possible: At least 2 weeks are recommended, I would recommend a month.

There is only one category of Schengen short-stay visa, namely, the type C short-stay visa. In applying, you must justify the purpose and conditions of the visit, which may be a business visit, a tourist visit, or indeed nearly any other kind of visit. In your case, of course, the justification is a business trip; in the case of your family members, it is to accompany you while you are on a business trip.

In countries that do have distinct categories of visa for business and touristic visitors, it is common to grant tourist visas to family members accompanying business travelers. It is also commonly recognized that business travelers will engage in touristic activities during their free time.

In short, your family members should just apply. If asked to categorize the purpose of the visit from a list of choices, then they can choose "tourism" unless there is another choice that describes their intentions more accurately.