Background
At the beginning of 1780, things were not going well for the British in North America. Burgoyne had been forced to surrender at Yorktown, Cornwallis and the Loyalists, despite beating General Gates at Camden, were doing badly in the Carolinas: Greene had taken over from the discomfited Gates, and, like Fabius against Hannibal, was, with his 'fight and run' tactics, forcing his opponents to tire themselves out with a series of Pyrrhic victories that led nowhere.
Opposition politicians at home were against the war, seeing justification in the American cause, and army and navy morale was suffering from lack of dedication to the war effort - many navy captains refused postings since they opposed the war.
Now France, Spain and the Dutch had joined in on the American side, not only affecting the balance on the mainland of North America, but also threatening British commerce and colonies elsewhere - notably in the Caribbean where the British felt vital interests were at stake (which they weren't on the mainland - the colonies, especially the northern ones, were rivals of Britain, not sources of wealth as the Caribbean ones were).
It was against this background that Admiral Rodney set sail for the Americas in January 1780 with 21 ships of the line. Off the Spanish coast, intercepted neutral merchantmen informed him that the Spanish fleet (which turned out to have eleven ships of the line) had left Cadiz. He sailed to meet them, in waters made even more famous 25 years later in 1805.
At one o'clock in the afternoon of January 16, the 74-gun Bedford, scouting ahead, fired a gun, loosed her topsails and signalled that the enemy was in sight.
The battle
In sight, but not brought to action. Outnumbered, the Spanish not unreasonably turned and headed back to Cadiz. Rodney, whatever the regulations said about forming line, simply ordered his ships to give chase as best they could. He also ordered them to engage from leeward (from the north-east) - effectively to overtake and get between the Spanish and safety.
Resolution, followed closely by Defence and Edgar, led the British in what had become a race, a race that the British won, though it was not until the sun was setting and a full moon rising that Resolution caught the trailing Spaniards.
The moon saw a subsequent mêlée of single-ship actions that lasted through most of the night. By its end, two Spanish battleships had run aground, one had been sunk, and one had escaped: the rest were in British hands. Rodney's flagship Sandwich had taken two ships herself - the Spanish flagship Fénix and, carrying on chasing, the smaller Monarcha.
Aftermath
The battle was not really of strategic significance - a victory over a minor opponent with the odds heavily favouring the British. The reason for its romantic place in naval history is that it was unique on two counts.
It was unique firstly in its abandonment by Rodney of any pretence of following the rules of engagement, even initially: no other fleet engagement was fought from the outset under the signal of 'General Chase'.
And even more importantly for the tellers of tales, it was fought at night under a full moon. So we know it as the 'Moonlight Battle'. Otherwise it might have been known, of course, as the first Battle of Trafalgar.
http://www.cleverley.org/navy/moonbattle.html
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