歓喜 means intense, almost triumphant joy or exultation. It is a more literary and dramatic word than the everyday 喜び (joy), making it especially memorable for learners.
In the song, 歓喜の仮面 means “a mask of exultation,” suggesting that even extreme happiness can be a role the singer wears to hide his loneliness and vulnerability.
"仮面" means "mask", and Janne Da Arc open by handing you a whole rack of them: "歓喜の仮面" (the mask of joy) worn "孤独を避けるため" (to avoid loneliness), "怒りの仮面" (the mask of anger) worn "弱い心守るため" (to protect a weak heart), and finally "笑顔の仮面" (the mask of a smile), right before the singer admits "僕はなぜか 泣いている" (for some reason, I'm crying). That is the twist of the song in one line: the happiest-looking mask is the one he cries behind. He lists his survival rules too, "声を殺して 耳をふさいで" (holding my voice back, covering my ears), and promises himself that if he just goes along with people he will "嫌われない" (not be disliked). Nice grammar tip here: "声を殺す" literally means "to kill your voice", but it just means keeping quiet so nobody hears you.
Then someone knocks: "胸の扉を叩いた" (knocked on the door of my heart). Their message is the heart of the song: "一人では何も出来ない" (alone, you can't do anything) is "'弱さ'じゃなく" (not 'weakness') but "'生きている意味'" (the meaning of being alive). The middle verse gets bleak, with his chest turned to "乾いた砂" (dry sand) and "息が出来ない でも外せない" (I can't breathe, but I can't take it off). By the end the other person does it for him: "仮面を外してくれた" (took the mask off for me). It is a bit earnest, sure, but the line "間違いは過ちじゃない" (a mistake isn't a sin) is a lovely thing to learn, because both words translate as "mistake" in a dictionary, and the song uses 過ち for the heavier, moral kind.