Barad-dûr was built by Sauron, the Dark Lord of Mordor, with the power of the One Ring, during the Second Age. The building took six hundred years to complete; it was the greatest fortress ever built since the Fall of Angband, and much of Sauron's personal power went into it.
Barad-dûr was besieged for seven years by the Last Alliance of Elves and Men, and was leveled after Sauron's defeat at the end of the Second Age, but because it was created using the power of the One Ring its foundations could not be destroyed completely unless the Ring itself should be destroyed. Isildur failed to destroy the Ring, and so the tower was re-built when Sauron returned to Mordor thousands of years later.
Only when the One Ring was destroyed did the Tower finally fall; without Sauron's power to sustain it, it could not stand. Barad-dûr collapsed to ruin and Sauron was finally defeated.
The Dark Tower was described as existing on a massive scale so large it was almost surreal, although Tolkien does not provide much detail beyond its size and immense strength. Since it had a "topmost tower" (the location of the Window of the Eye, from which the Eye of Sauron gazed out over Middle-earth), it presumably had multiple towers. It is otherwise described as dark and surrounded in shadow, so that it could not be clearly seen.
Barad-dûr in the movies
In the Lord of the Rings movies by Peter Jackson, Richard Taylor and his design team built a 9 foot high miniature ("big-ature") of Barad-dûr for use in the film. Using the size scale for the model implemented for the films, the Dark Tower is depicted as being over 1,500 meters (5,000 feet) tall, three times as tall as the Sears Tower in Chicago.
The Return of the King film also shows Barad-dûr as clearly visible from the Black Gate of Mordor. Even granting its enormous size, it was located one hundred miles away and to the east of the Gate¹, not to mention being behind the inner mountain ridges of Udûn so Aragorn's army would probably not have been able to see it. In the film version, the geography of Mordor seems generally to have been compressed somewhat, perhaps for artistic reasons related to rendering such complex stories in a visual medium. In the case of the Black Gate scene, having Barad-dûr visible from the Gate means that the army can see the Eye of Sauron staring at them.