Att supa is a verb that means 'to drink alcohol', usually with the connotation of drinking to excess, with the goal of getting drunk. You use it without a noun object, so in conjugated form, you say 'han super' and not 'han super alkohol', but you'll also hear the phrase 'att supa sig full' which literally means 'to drink oneself drunk'. 'En supare' is a (heavy) drinker.
But the word has much more innocent origins, shared with the English word 'soup' and its Swedish equivalent soppa.
In Old Norse and Old Swedish, supa meant 'to drink' and was often used in particular for foods that were consumed in liquid form rather than beverages. One recipe that was popular from the 16th century onwards was ölsupa (beer soup), in which the alcoholic drink was mixed with milk, flour, and sometimes eggs or sugar.
So the verb att supa was used to mean 'to eat soup', but because of the popularity of beer soup, the term gradually came to include other alcoholic drinks too: first, those which were consumed with a spoon, and later others as well. Swedish also has the noun 'en sup' meaning 'a sip' but used only with alcoholic drinks, in phrases such as 'att ta en sup' (to take a sip). There are equivalent verbs in Danish and German with the same origin: zuipen and saufen.
The related word att insupa has a neutral meaning, equivalent to 'to consume', but this has become more generalized, also meaning 'to breathe in' or 'to absorb', so you can insupa an atmosphere.
There’s also the term kålsupare (literally ‘cabbage drinker’) which isn’t used much on its own, but comes as part of the idiom ‘lika goda kålsupare’ (equally good cabbage drinkers). You can use this phrase to demonstrate that two people are equally useless or otherwise alike in a negative way. In English, it could be translated as ‘birds of a feather’ or ‘both as bad as each other’.
Jag vill inte supa
I don't want to drink (too much)
Vi ska supa oss under bordet
We'll drink ourselves under the table (part of a typical Swedish drinking song)