Grog is "One part sour, two parts sweet, three parts strong, four parts weak," i.e. lime juice, sugar, rum, and water.
Rum is associated with pirates because it is made from sugar cane which is common in the Caribbean. The golden age of piracy occurred primarily in the Carribbean and eastern pacific ocean, and sugar and rum were produced and exported in the colonies in these areas. Piracy was rampant at the time, and so piracy is associated with rum, as opposed to other alcoholic drinks which would not have been widespread or available at the time.
"Grogram" is a type of heavy stiff cloth (the name derives from the French grosgrain), and in the early 19th century a certain British Admiral named Edward Vernon became known among his sailors for always wearing a coat of grogram. This earned him the nickname of "Old Grog", and when he came up with the idea of mixing the rum ration with water and lime to keep it fresher and combat scurvy, the resultant drink took the name Grog.
As an interesting aside, Vernon's Wikipedia page also notes that George Washington's home of Mount Vernon is named after Admiral Vernon - Washington's elder brother served in the Royal Navy under Vernon.
There are a host of recipes for grog, but in the British navy during the period I study, rum was issued at a half pint per day in the morning and evening, mixed as "three-water" grog (three parts of water to one of spirits). I wrote about this somewhat more extensively in this question:
Interestingly, we tend to associate rum with sailors (possibly due to pop-culture associations with pirates), but the British navy did not have an official rum ration until 1844. Before that, sailors were entitled to some form of alcohol, but it varied by station (sailors in the Channel Fleet would get a gallon of beer per day; sailors in the Mediterranean would get wine; sailors on the Copenhagen expeditions apparently drank akavit, etc.) Rum was originally for service in the Americas, but it became useful for all sorts of foreign service, because it stows much smaller than beer.
The questions about its specific association with piracy are fairly on topic, in the sense that we think of "pirates" (the kind with eyepatches and parrots) in the West Indies, where rum was made. But in terms of why the pop culture association of pirates and rum has been made, /u/davidaop is your go-to.
Edit: I should mention that the practice of mixing lime into grog did not become regularized until the latter half of the 19th century -- before that, ships' captains and pursers had a wide latitude for what they could use as an antiscorbutic.
Grog is watered down rum.
Most rum comes from the Caribbean. The Caribbean itself is heavily associated with Pirates (think a certain popular Disney ride and movie franchise). So pirates, often found in the Caribbean, became associated with rum, an alcohol prevalent in the region.