A swig is a big, fast gulp, like the swig you take from your water bottle after hiking up a hill on a hot summer day.
If your dainty friend is delicately sipping her iced tea while you swig yours, you'll definitely be finished first. To swig is to take a big, healthy slurp of a drink. Originally this informal word was associated with alcoholic beverages, but today you can use it for anything you're drinking with gusto. Experts aren't sure where swig comes from, but some guess it's connected to the Old English swelgan, "to swallow."