Which military battle in history do you think was the most famous or had the greatest impact?

I will limit my response to American History since the list of world events are too plentiful to pull out a single battle.

The Battle of Valverde near Socorro, New Mexico during the American Civil War is not well known, but it marked a serious blow to the Confederacy's potential for winning the war. The battle was fought between Confederate volunteers (with professional officers) and a small number of Union regulars with New Mexico volunteers (led by the legendary western figure, Kit Carson).

If the Union had lost this battle then it is likely that the Confederacy would have won at Glorieta Pass (near Las Vegas, NM) and moved on to Colorado riding a wave of momentum. Colorado would have fallen, then Utah followed by the remainder of Arizona (Tucson and the southern portion of AZ had already been taken by the Confederacy). With the Southwest conquered, it would have been an easy march into California and the ports of the West. These ports would have allowed the Confederacy to circumvent the Union blockade in the East and provided international legitimacy to the Confederacy.

Fortunately, the Confederate leadership (General Sibley) was incompetent, the New Mexican terrain was too difficult, and Colorado volunteers destroyed any chance the Confederacy had in the Southwest at Glorieta Pass.

Many will disagree, but this is my evaluation.

The Battle of Mogadishu - largest firefight since Vietnam and painfully taught us, once again, that despite possessing technological superiority, we can never underestimate the resolve of a rag-tag, drug induced, seemingly disorganized mob defending its home turf.

The Battle of Salamis was fought between an Alliance of Greek city-states and the Persian Empire in 480 BC, in the straits between the mainland and Salamis, an island near Athens.

No doubt most have seen the movie 300 if they've had no other exposure to these events; to block the Persian advance in their bid to conquer Greece , a small force of Greeks blocked the pass of Thermopylae, while an Athenian-dominated Allied navy engaged the Persian fleet in the nearby straits of Artemisium. In the resulting Battle of Thermopylae, the rearguard of the Greek force was annihilated, whilst in the Battle of Artemisium the Greeks had heavy losses and retreated after the loss at Thermopylae. This allowed the Persians to conquer Boeotia and Attica. The allied Greeks prepared to defend the Isthmus of Corinth while the fleet was withdrawn to nearby Salamis Island.

Although heavily outnumbered, the Greek Allies were persuaded by the Athenian general Themistocles to bring the Persian fleet to battle again, in the hope that a victory would prevent naval operations against the Peloponessus. The Persian king Xerxes was also anxious for a decisive battle. As a result of subterfuge on the part of Themistocles, mirroring the betrayal by Ephialtes at Thermopylae, he sent an informer, a slave named Sicinnus, to Xerxes to make the Persian king believe that the Greeks had in fact not been able to agree on a location for battle, and would be stealthily retreating during the night, the Persian navy sailed into the Straits of Salamis and tried to block both entrances. In the cramped conditions of the Straits the great Persian numbers were an active hindrance, as ships struggled to maneuver and became disorganized. Seizing the opportunity, the Greek fleet formed in line and scored a decisive victory.

Xerxes then retreated to Asia with much of his army, leaving Mardonius to complete the conquest of Greece which he failed to do. Afterwards the Persians made no more attempts to conquer the Greek mainland.

A Persian victory would have hamstrung the development of Ancient Greece, and by extension western civilization, Salamis is thus the most significant battle in human history.