I've recently gotten an invitation letter for a conference, in Germany, for a week in September. I wish to stay two extra weeks for sightseeing/leisure. What documents do I need for my visa application to allow them to extend my stay for two more weeks?
there is mention of a hotel reservation document that I would need to submit. How to put that I will be staying with family?
Its not really hotel reservation, what is required is proof of accommodation so the family you are staying with can provide a letter stating that you will be staying with them and the period of your stay.
Regarding your bigger question; you are basically going for a trip, a part of which will be to attend a conference, and then the rest is for tourism.
You need to provide an itinerary that covers these scenarios (for example, your flight reservation should match the dates you intend to stay in the area).
In the application, make sure you write the correct number of days you'll be staying in the area, I usually calculate it like this:
1 day before the conference + ___ days of the conference + 1 day after the conference.
So for a 7 day conference, I plan 9 days (this is to count for any travel/flight issues).
Next, add the number of days you plan to remain after the conference, which is 14 days. So your total duration is 22 days (1 day before conference + 7 days for conference + 14 days for tourism + 1 day after buffer).
This is what you should fill in the Schengen form in the "Duration of Stay" box (#25 if my memory serves me right).
This is to make sure you aren't accidentally given a visa for the duration of the conference only. Schengen visas should be valid for the entire duration of the stay.
Next, you'll need to prove that you have sufficient funds available to you for the duration of the stay. The exact amount differs from country to country; but on average 60 EUR per day is a comfortable amount. This is money that you should have accessible to you (either in your bank, or via credit cards, etc.)
Finally, I would provide an itinerary for the portion of your stay outside of the conference; because one can do many things in 14 days and (especially if this is your first application) they will want to know what exactly you are planning to do once the conference is over.
Of course, this is just guidance - the decision is up to the counselor officer that judges your application. Don't forget the normal documents you need for a such a trip (insurance, tickets, proof of ties, etc. etc.)