The band were working class cockneys from North London, and had known each other since childhood; sang with their natural English accents, and often about the events of ordinary life.
The musical style was - rhythmically - heavily influenced by Jamaican Ska and Rocksteady reggae (their name came from a track by Prince Buster and that is who is referenced by the first track below: The Prince); but they put an unique stamp on it, and used careful and effective arrangements (for which I think the keyboardist, Mike Barson, was mostly responsible). It often had the quality of a fairground organ.
As musicians, they were excellent. The lead singer - Suggs - had a distinctive and very natural way of putting across a lyric. And the seventh member of the band - Chas Smash - provided much of their special visual identity with his original (and effective) style of dancing and fashion (see videos below, most clearly on Baggy Trousers), backing vocals; and his interjections and interactions with Suggs.
Indeed, the brilliance of Madness was very much visual, with a trademark way of marching in-line and in-step, as a kind of human caterpillar (see illustration above) which featured in most of their videos); and various other tropes such as a manic, bared-teeth, fixed 'smile' - and particular ways of moving, interacting - as well as dancing.
In general, a surreal - hit and miss, semi-private, take-it-or-leave it - style of humour - which they called 'nutty'.
All very different, and English - and, although something of a British 'institution' - the band never made significant popular headway in the USA.
But best and most important was the music, some of which has a permanent appeal to me: