A 'colloquy' is a formal word for 'conversation', so colloquial language means the everyday language or register we adopt when chatting to friends, for example, e.g. 'Hello Fred, how's the new mother-in-law these days?'. Slang is a particular form of colloquial language used by certain social groups, e.g. 'Hey-up Fred! How's the new battle-axe then?'; 'Hey that's some cool dude there!'