what is big red wine?

Likely they mean full bodied.

Cab. Sauvignon, chateneuf de pape, etc, pick at French or Italian. Most American/ Australian wines are not big enough.

Sangiovese, Primitivo, Nero d'Avolo would all work well. they drink well, too,

For braising beef in wine, Chianti also works well and is easy to find (I use it often for stracotto - Italian "pot roast.") Should be a drinkable one, but need not be on the high end of the cost spectrum.

Absolutely, the wine should be drinkable, but don't use the most expensive bottle you can find--Chianti and the other wines I mentioned are available at reasonable prices.