The mask is worn by unnamed white glove society workers. It is stated in-game that the intention of wearing the masks is to create an atmosphere of mystery with regards to the patrons' impression of the hotel.
Equipping this item singularly does not create a pop-up notification that the character is dressed as a member of the White Glove Society. However, un-equipping it when worn without the other component notifies one that they are no longer disguised as a White Glove member.
When given to Veronica, she will equip the mask and then offer to teach you a new unarmed attack move as if you gave her a dress.
A different version of the mask is depicted on the Collector's Edition playing card, lacking the gold trim and covering the wearer's entire face.
One of the most useful bugs in Fallout: New Vegas includes the simultaneous use of the White Glove Society attire and the White Glove Society mask. In fact, you can get idolized with almost every faction. You must wear the attire first, then put on the mask. Now don't take off the attire, but equip a faction outfit over it. After this remove everything. If everything has been done correctly, check out your faction relationship and you should be idolized by almost everyone. [verified]
There is a bug where the disguise is not removed with the mask, permanently setting your reputation with several, high-profile factions to neutral. Possible causes and solutions are available here. [verified]
Collector's Edition playing card with the mask
↑The Courier: "Why is everyone around here dressed up and wearing masks?" Chauncey: "Those are all members of the White Glove Society. Our founder, Marjorie, gave us all a dress code. There's only one rule to it. In her words, we must dress in such a way that no one can be said to have dressed better than us. As for the masks, I'm not allowed to tell you. We're sworn to secrecy. Actually that's not true. That's just what we're supposed to say. I think Marjorie likes them for the mystery they create. And the way they make it clear that we're different from everyone else. But you didn't hear that from me." (Chauncey's dialogue)