The residency format
Rather than a single tour date, a musical residency has an artist performing recurring dates at one Vegas venue over weeks, months or years — the same structure that supports the city's Cirque du Soleil and magic/comedy residencies, applied to major recording artists.
Why artists do it
A residency lets an artist build a show around one stage's specific technical capabilities in a way a touring production can't, and gives fans a reliable window to plan a trip around rather than chasing a single tour date.
Checking what's currently running
Residency calendars change over time as artists' runs begin and end — check current listings directly if seeing a specific artist is the reason for your trip, rather than assuming a past residency is still active.