Just a short version: It was founded by British Airways to compete with LH in Germany, focusing especially on inter-German travel. They failed in almost every respect, but became very popular by their passengers. However they were NEVER profitable. So in the end Rudolf Wöhrl took over (in 2003 or 2004) the DBA and concentrated on German routes. Since British Airways was out, it was called DBA now, keeping the name but removing any reseblance to British Airways.
Surprisingly he managed to get DBA profitable again.
DBA is very popular. They focus on frequency, offer a snack and a drink on their flights for free and have no extreme low fares, but usually the best offer. You don't have to search a lot to find a cheap flight with DBA. So DBA is great in terms of reliability and custumer service.
It should be noted, btw that the airline always was German. Their management is German, they have a employee representation (Betriebsrat), so they are very conventional, which is a nice contrast to Air Berlin.
To me there always was a big question mark behind the DBA. They have an old fleet (but just ordered new 737NGs), seem to have no alliance and focus on German flights. But maybe that's their secret. Many business travellers like DBA, and in a recent test by Stiftung Warentest, they were rated to be one of the best LCCs (better than Easy Jet, Germanwings, Ryanair, HLX), because their product was extremely reliable, reasonably cheap and had a lot of free baggage.
They have a lot of custumers who fly them regularly. So I hope they will stay.