Pieta is both a specific statue and a series of marble statues sculpted by Michelangelo. Every Pieta includes the Virgin Mary and Christ, and they sometimes also show other religious figures.
There are four carvings that may be called the Pieta, although the last two were never completed, and are not in very good condition. But when referring to 'the Pieta' people usually mean the one housed at St. Peter's in Rome, which depicts Mary holding the dead Christ. This is by far the best of the four--take a look at it here.
Aside from being a great work of art, this is the only surviving piece of work signed by Michelangelo. I believe that this one is his first Pieta, although it is very hard to find information on the correct chronological order of the statues.
Next comes the Pieta at Florence, santa Mria del Fior. This one was intended to be put over Michelangelo's tomb. Christ is being supported by Nicodemus from behind, Mary Magdalene on his right, and the Virgin Mary on his left.
While the first one is in near perfect condition*, and the second one is pretty good, the Palestrina Pieta in the Academy of Fine Arts of Florence is rather worn. Christ is in good enough condition, but the Virgin Mary, holding him up from behind, looks like a picasso. Magdalene, at his side, looks a bit better, but she's a little fuzzy around the edges.
The last one is the worst, by far. The Rondanini Pieta, located in Milan, in the Sforza Castle Civic Museums, looks as though a beaver has used it as a chew toy. The Virgin Mary is holding up a human shape that must be Christ. As far as I can find, it is most likely that the damage came from the centuries it stood in the courtyard of the Palazzo Rondanini. You can see it here.
Pietà is an Italian word that is usually translated (in this context) to mean mourning. It's interesting to note that a common translation of Pietà is also pity, which shares a Latin root (pietās) with piety. I suspect that there is a double meaning going on here, but I do not know enough of 15th century Italian to say anything definite.
These days pieta may refer to any work (painting or sculpture) depicting the Virgin Mary holding the body of the dead Jesus, although unless it is clear from context which pieta is being referred to, you can assume that it is the one at St. Peter's.
* The St. Peter's Pieta was attacked by a madman named Laszlo Toth in 1972. He knocked off Mary's nose, part of her eyelid, and her right arm at the elbow. All has been made right again by the Vatican art-restoration laboratories.