Oy, oy, oy! First there is the question of spelling. Is it a bencher, a bentcher, or a bentscher? Answer: I’ve seen all three.
And no, it isn’t a piece of furniture, although that’s what it sounds like.

A bentcher is a little book or folder with the text of the blessings said after a meal, the birkat hamazon. It comes from the Yiddish word bentch, which means “to bless.” (Thanks to both Anne and Jeff, readers who corrected me on this.)
Bentch – to say or sing the birkat hamazon (blessing after meals.) Some may say, “It’s time to bentch,” meaning, the meal is over already, let’s bless and be done!
Bentch gomel – to say a blessing of thanks for delivery from danger. Always said during the Torah service.
Bentch lulav – to say the blessings that go with waving the lulav.
Bentcher is the book with the birkat hamazon in it. In a household where they bentch after every meal, it will likely be one of a half-dozen stuck in a napkin holder on the table. Some bentchers also have zmirot (zmee-ROTE) which are Shabbat songs for the table.
Some Jews carry a mini-bentcher that folds up to credit card size, to use when eating away from home.
Oh, and in case you are wondering, Hebrew for “blessing” is brakhah.