Cortez, with less than 1,000 Spanish, and perhaps 3,000 allies, was facing tens of thousands of Aztecs led by a brother of the new ruler.

(All numbers involved in the battle are much disputed estimates, but Cortez was undoubtedly massively outnumbered).

Cortez' main advantage was that he had cavalry (no more than a hundred and probably less) on a flat battlefield ideal for cavalry, against an opponent totally unfamiliar with cavalry.

The other advantage was that Cortez directed his cavalry at the Aztec commanders, theorizing that if he could slay the leaders, the massive army opposing him might crumble. This worked.

Cortez himself seems to have led the cavalry charge against the Aztec command, but he did not personally slay the leader:

Diaz del Castillo tells it:

"When Cortes saw [Matlatzincatl] with many other Mexican Chieftains all wearing great plumes, he said to our Captains: Now Señores, let us break through them and leave none of them unwounded” … and the horsemen charged, and Cortes struck his horse against the Mexican Captain, which made him drop his banner, and the rest of our Captains succeeded in breaking through the squadron which consisted of many Indians following the Captain who carried the banner, who nevertheless had not fallen at the shock that Cortes had given him, and it was Juan de Salamanca… who gave him a lance thrust."

This battle (after Cortez had broken free from the Aztec capital) was perhaps the last great chance for the Aztecs to crush Cortez' small remaining force and possibly alter the course of history. They failed. Cortez won the battle, rallied the allies, and eventually conquered Mexico.

For a much more detailed account, see the excellent previous response to a similar question by /u/400-rabbits, here:

http://www.reddit.com/r/AskHistorians/comments/1pmwda/why_did_the_aztecs_lose_the_battle_of_otumba/