I very much agree with Sabine's recommendations of San Clemente ( for quality and value) Abastos for new and innovative cooking ( although a bit portion challenged for me) and The Obispo on the Rua Franco for tapas.
Add to that list the Securusal in the Plaza San Agustín which will also cook fresh fish and seafood you buy at the market and the Garum for an excellent and varied Menu del Dia and fantastic tapas as well as extensive wines and craft beers.
In the Old Town near Ivar's place the Plaza Mazarelos is worth visiting - the only surviving city gate stands there - through which the city's wine supplies were delivered. There are good terrace restaurants there and the relatively new Puerta do Vino restaurant which is (slightly) more up market.
For fun ( if you don't mind the expense) the Michelin starred Casa Marcelo in the Rua Hortas just off the Obradoiro is worth a visit. And a recent favourite is the Restaurant Hortas opposite which boasts a stunning garden at the back plus modern Galician cuisine cooked by two young chefs who imho have a great future.
In the Rua San Pedro near where the
Camino Frances enters the Old Town there is an abundance - Dieciséis has been mentioned and is consistently good and the new neighbouring restaurant A Moa is just excellent - again with a lovely garden. Nearby is one of Santiago's Japanese restaurants if you fancy Sushi.
For that special meal, the restaurant Pedro Roca ( 100 metres from the Pilgrims' Office) is outstanding - Chef Patrón Pedro is very talented and the menu has everything you could want in modern five star cooking.
For an adventure either walk 20 minutes or get a taxi to the O Almacén restaurant which does fantastic fish and meat in enormous portions on a huge wood fired grill.
Hungry yet?
Oh I almost forgot... In the Rua da Porta da Pena the nondescript entrance to the Casa Felisa leads through the bar to a gorgeous garden restaurant. On weekdays they have a Menu del Día as well as a la carte. Well worth a visit.