Connie isn't quite a major character—but she's not quite minor either. She's fairly important. The story begins at her wedding after all. However, that initial pleasant experience quickly descends into hell.
Her husband, Carlo, is viciously abusive—he beats her in one of The Godfather's most unpleasant scenes. Despite this, she's also deeply upset when Michael has Carlo murdered. She accuses him of evil hypocrisy for becoming the godfather of her child, while having his enemies slaughtered at the same time.
The role is played by Talia Shire, Francis Ford Coppola's sister.