Aside from a few minor foreign exceptions (Dutch in Suriname, English in the Falkland Islands and Guyana, and French in French Guyana), one major exception (Portuguese in Brazil), and countless indigenous languages (Quechua up in the Andes or Tehuelche in Patagonia), the primary spoken language in South America is Spanish.
If you speak some Spanish, you'll notice distinct regional differences – from Peru to Chile, Argentina and even Uruguay: each country has its own peculiarities regarding vocabulary and pronunciation so you'll no doubt find conversations with locals to be both amusing and enlightening!
Yet, as far as travellers are concerned, it's important to stress that you can travel through all of South America without speaking any Spanish or Portuguese at all. Would it help to know a few local greetings and phrases and, perhaps, have a basic grasp of numbers? Sure! Locals love it when you 'attempt' to say at least 'hola!" However, speaking Spanish or Portuguese is no prerequisite to having a fantastic time on your South America tour!