The Matterhorn is one of the very few peaks whose fame has trancended the realm of the mountains; perhaps only Mount Everest has entered the public consciousness more than this sharp summit of the Swiss-Italian border.
The renown of the Matterhorn is not due to its elevation. Depending on the prominence threshold selected, it could be called anything from the fifth to tenth highest peak in the Alps. Only 14 km/9 mi away is Liskamm, a slightly higher summit, and just beyond that is the Monte Rosa massif, the second highest in the Alps, rising 156 m/512 ft above the Matterhorn's crest.
The Matterhorn is famous for two reasons. The first and most obvious is its shape, especially in the classic view from Zermatt. A tall rocky fang with a summit that appears to overhang the sheer north face, the first view of the peak is breathtaking; it does not look like any other mountain, but instead a surreal landform that just looks impossible, set among the more mundane summits around it.
Aside from its profile, the Matterhorn is legendary as the site of perhaps the most celebrated mountain climb and accident ever. Edward Whymper, on his seventh attempt, climbed this last major virgin alpine peak on July 14th, 1865, only to have four members of his seven-man team plunge to their deaths on the descent. Whymper and his two guides were saved only when their rope broke.
These days, the Matterhorn is a relatively easy and very popular ascent. I have heard that the Hornli Ridge (Whymper's route and still considered the easiest) on nice weather days is a solid queue of climbers, and that the greatest dangers are people-generated rockfall from above and tricky maneuvering while passing other parties going opposite directions. It is not a climb for the acrophobic, but a guide can take any competent scrambler to the summit without much difficulty. Other routes are harder but less crowded, and the north face remains one of the "big three" walls of the Alps, along with the north faces of the Eiger and the Grandes Jorasses.