To drink in big mouthfuls, usually straight from a bottle or can, without being careful or polite about it. The picture is someone tipping a bottle up and gulping — a worker finishing a bottle of water after a hot job, a friend taking a drink from your bottle at a night market. It is informal and a bit rough round the edges, so it fits friends and family, not a business report. Two things are inside the word: the size of the mouthful (big) and the manner (casual, thirsty, unrefined). Because of that casual feel, it often comes up with beer and other alcohol, and English speakers sometimes use it to hint that someone is drinking too freely — 'he was swigging whisky all evening'. But there is nothing about alcohol in the word itself; you can swig water or juice perfectly happily. It is also a noun: a swig is one such mouthful, and 'take a swig of' is the most common way you will hear it.