Our two leads are Winfried Conradi (played by Peter Simonischek) and his daughter Ines Conradi (Sandra Hüller). Winfried is an inveterate prankster, making people uncomfortable by making bizarre statements without clarifying if they're a joke, or wearing costumes or a set of false teeth. Ines is a manager temporarily in Bucharest, trying to outsource oil industry jobs. She's bitter and unhappy, has a crappy boyfriend and friends she's not really keen on, but doesn't seem to realize any of it. Her father - who she has very little time for - spontaneously visits her in Bucharest. On seeing what a mess she is, he starts to prank her - which doesn't really help. But then, one night, he drops in on her and her friends at a bar with his bad teeth and a fake wig. And his daughter doesn't shut him down. So he builds a persona around the name he invented ("Toni Erdmann") and - with her implicit agreement - spends more time among her friends and business associates, and kind of turns her life upside down.
I don't like Winfried's style of humour, his love of making people uncomfortable. I was kind of put off by that. And the movie is too long (162 minutes), but it does put it to good use, to let you know that Winfried really loves his daughter and is, in his own incredibly unorthodox way, trying to help her. The naked party near the end of the movie is both incredibly cringe-inducing and hilariously funny - why it happens would be too much of a spoiler. You'll have to decide for yourself if this is your style of humour: it's not slapstick and there's a real story there, but be prepared to cringe a few times.