Try this
SELECT M.FirstName, M.LastName
FROM Members As M
Join xrefartistsmembers As X On X.MemberId = M.MemberId
Join Artists As A On A.ArtistId = X.ArtistId
WHERE A.ArtistName = 'Today'
SELECT FirstName, LastName
FROM Members As M
Join xrefartistsmembers As X On X.MemberId=M.MemberId
WHERE X.ArtistId = 16
In the sample data you've provided, there's no data for 'Today'.
So, to get some realistic data with the data you provided, below query may give you more clarity
SELECT M.FirstName, M.LastName
FROM Members As M
Join xrefartistsmembers As X On X.MemberId = M.MemberId
Join Artists As A On A.ArtistId = X.ArtistId
WHERE A.ArtistName = 'Confused'
FirstName LastName
Doug Finney
Terry Irving
Michelle Henderson