Clashed dies occured when dies were damaged by striking each other without a planchet between them, imparting part of the obverse image to the reverse die and vice versa. Most examples are minor, having no profound affect on the value of the coin other than numismatic interest. They're very common on Indian cents and usually more clearly seen on their reverse due to the larger field (see the 1877 page of the "Rarity by Date" section of this website).
Those with special detail and location can receive variety status, like the 1880 S1 imaged below. The obverse rim of the die, complete with inverted denticles and "ICA" of AMERICA, is seen on the reverse of this coin in marvelous detail. The fact that it found its way this far into the center of the coin, resulting in a rare offset clash, makes it even more extraordinary.