Auerbach‘s Keller 
The second-oldest restaurant in Leipzig and famous for its association with Goethe, Germany's equivalent of Shakespeare. One of the scenes of Goethe's best-known work Faust is set in this vaulted, basement-level restaurant. I ate there quite some years ago and remember it as serving good wine and filling German food.

Bayerischer Bahnhof
I and two fellow Union Berlin fans (a Scot and an American) ate here prior to Union's cup semi-final in the city a couple of years back. They offer beer brewed on site and hearty German fare. It's located a little way from the city centre, but is still within walking distance of the main station and city centre hotels.

Ratskeller
I have to admit that I'm a sucker for German town-hall cellar restaurants! I love the vaulted ceilings, the oak furniture and the hearty fare. Not to mention the odd glass of beer or fine wine. Like many German towns, Leipzig too has a traditional restaurant in the basement (Keller) of its town hall (Rathaus), thus the name Ratskeller. As the website linked to above is all in German, here's the Ratskeller Leipzig page on Tripadvisor.