It is basically performing with a live band behind them rather than a backing track. Technically it could mean that they do old Rat Pack/Swing style music but they rarely do.
Michael Buble is the king of performing with a big band in recent years. He flows with the band rather than trying to sing over it, and the fact his voice is beyond powerful also helps
Also for anyone who has seen him live, his finale for each show is usually Accapella when the big band stops playing but he also drops his mic and just sings without one and the whole arena goes quiet to listen to him fill a 15,000-20,000 seat arena without a mic. Quite a sight to see.