A chronological retrospective of the two rings Louis wears. For 200 years. Apparently Lestat could get him to change his wardrobe, kill people, and all kinds of other depraved things etc., but Louis kept his jewelry to a strict minimum.

One appears to be his wedding ring.

• in the movie version, Louis is a widow. Maybe the ring was his wedding ring.
• After the Dark Gift, he keeps it on. Maybe it’s earned new significance towards Lestat, now. 
• After Lestat’s “death,” Louis continues to wear the ring. Maybe it’s significant to both people he’s lost.

The other is a pinky ring.

• This one may have just been fashionable to begin with.
• Maybe it symbolized the child who died with his wife in childbirth.
• Later, maybe it’s earned new significance when Claudia is given the Dark Gift.
• After Claudia’s death, Louis continues to wear the ring. Maybe it’s significant to both “children” he’s lost.

UPDATE: anaryawe says:

(re: Louis & rings in the film) pinky ring was a gay code, or “no marriage plz”-code at some point. I’ve always thought that Louis’ pinky ring is his dead wifes wedding ring, and so signifies both “all the loves I’ve lost” and “tried heterosexuality once and it was a disaster”…