When talking about physical size, we use big for most cases. Large is a little more formal.
Everyday: Elephants are big animals.
In a textbook: Elephants are large mammals found in Africa and Asia.
In informal English, we use "large" for size when talking about clothes and drinks:
I ordered a big coffee with milk.
I ordered a large coffee with milk.
Sizes for shirts, socks, and other clothing items are small, medium, and large. The word big is often used in collocations with a happening or event, for example:
a big accomplishment
a big decision
a big disappointment
a big failure
a big improvement
a big mistake
a big surprise
The word large is often used in collocations involving numbers and measurements.