Short for 'traditional', and in music it names traditional jazz — the older New Orleans style that was popular in the 1920s, where one instrument (usually the clarinet, or the trumpet leading) runs a busy, decorated line while the others hold a steady beat underneath. It is especially a British word: in 1950s and 60s Britain there was a whole 'trad jazz' revival, and 'trad' became the everyday name for that sound. You will also hear 'trad' in British folk music, meaning a song so old that nobody owns it — the credit on the record just reads 'Trad.' More broadly, and this is worth knowing because you'll meet it online, people now use 'trad' as a casual adjective for anything old-fashioned or done the old way, sometimes warmly and sometimes as a joke; in the last few years it has also picked up a political flavour in some corners of the internet, meaning traditional in a social or religious sense. The music sense is the settled, respectable one.