Period pieces are stories that take place in the past, and/or are strongly evocative of the past not only in their settings, but in dress, character attitudes, and often story telling devices. They frequently invite have lavish production values spent on art direction and costumes. They are often dramas or comedies of manners, and frequently are adaptations of literature that would lose something in simply having the story and characters updated to the modern day (though both storytelling approaches have their merits—see Clueless versus the original Emma), and also often feature large, sweeping romances.
Very often Oscar Bait focusing on a few key characters and letting actors flex their dramatic muscles with grand speeches, declarations of love, and other impassioned monologues. For literature, there are certain authors/stories that get recycled, no matter how many times they've been done before. (Adaptations of the Brontë sisters, Jane Austen, or Charles Dickens are especially common.) Useful if the period predates a Trope Breaker.
The BBC makes a lot of these, and PBS imports them wholesale. The BBC have sometimes even been accused of "falling back" on costume dramas when they've gotten into trouble for something or when the licence renewal is coming up and they need to show they can do some "proper" drama. That said, they're very good at making them: they usually garner a lot of praise, and the end result is packaged into box sets for sale during PBS pledge drives. As a side effect of their decades of period piece production, The BBC has accumulated a vast stockpile of costumes and expertise, which is occasionally raided by sister shows such as Doctor Who when they decide to do a historical episode.
Compare Historical Fiction. Often overlaps with Costume Drama. Contrast Present Day and The Future. Compare and contrast 20 Minutes into the Past, where the difference is so minor that it's barely noticeable. See also Unintentional Period Piece, which becomes one by accident due to capturing key moments of the era in which it was produced, and * Historical Re-Creation, which is a period piece-themed Reality Show.
Period pieces often feature bits of setting, characterization, and plot that could not be updated to the modern day without requiring significant changes. For example, setting Pride and Prejudice in the Present Day as is would make the crisis (Wickham's eloping with Lydia) seem inconsequential and the reaction silly, since the entire point of the crisis is that if Lydia doesn't marry Wickham and clear the family name, the rest of the sisters will end up as impoverished old maids because no gentleman will marry them. That's the way life was in 1812. (Even further proved that, in The Lizzie Bennet Diaries, they had to change Lydia's scandal to make the plotline work). Technology Marches On can also lead to a period setting; in Super 8 the titular media's three-day minimum turnaround time was a load-bearing Plot Point that meant the characters working in instantly-viewable digital video simply wouldn't serve the narrative.
On the other end of the scale, if the story would make more sense set in the present because the characters exhibit modern values that's Purely Aesthetic Era; if it's set around a specific event in the recent past, enough that the lack of distance from the era leads to sloppy set mistakes it's Present-Day Past.