There were many famous black singers/musicians, such as Louis Armstrong, Ella Fitzgerald, and Nat King Cole. How were they treated back then? Did their race affect their careers?
Las Vegas was an interesting case. It used to be known as the "Mississippi of the West." The hotels on the Strip were segregated. Black entertainers would fill up the ballrooms, but then they'd have to enter the properties through the kitchens. There was one female black singer whose name I can't recall who demanded to be allowed to swim in the swimming pool. They cordoned off the pool so she could swim alone, and after she was done they drained and refilled it.
The Rat Pack did a lot to change this. The Moulin Rouge was an integrated casino that opened up in the Westside, which was the black side of town (now it's the 'traditional' black side of town). Its run was short but legendary. Basically, the Rat Pack would do a show at the Riviera, and at 10 p.m. when the show was over they and their entourage would pack it up and head on up to the Moulin Rouge to party until dawn. All the cool people would follow them, and it was making the Strip hotels look really bad. The big hotels were desegregated before long.
As to why Vegas was so segregated: 1) it was a railroad town before it was a tourist town (there was nothing but a couple of ranches until 1910 when the Southern Pacific went through), and railroad bosses from the South ran things, implementing segregationist policies early on (this also has something to do with the "Rebel" mascot at UNLV, which was more blatantly Confederate in decades past, whereas now he's more of a frontiersman); 2) the big hotels were paranoid about alienating Southern tourists.
This Smithsonian Magazine article gives a pretty good overview.
In segregated areas of the country, black celebrities were treated just like all other black people--only allowed to use segregated facilities, barred from whites-only hotels, that kind of thing. The Jesse Owens movie, Race, is coming out this weekend. It showcases how Owens embarrassed the Nazis during the 1936 Olympics. However, when Owens came back to the US, he, by his own admission, was treated worse than by the Nazis. He couldn't stay in the same hotels with the other Olympians. President Roosevelt invited all of the Olympic athletes to meet him after the games... but Owens was excluded.
Likewise there was a big controversy when Marian Anderson was supposed to sing in Constitution Hall, and the Daughters of the American Revolution, who controlled the hall, wouldn't let her perform there for a non-segregated audience. In that case, Eleanor Roosevelt, who was very concerned with Civil Rights, stepped in and arranged for Anderson to perform at the Lincoln Memorial, to a much larger audience and more prominent publicity than she would have had at Constitution Hall.
All black performers faced treatment like this at that time.
I assume you're just referring to how they were treated in the USA?