This page contains information regarding Star Trek: Picard, and thus may contain spoilers.
(covers information from several alternate timelines)
Romulan ale was a highly intoxicating alcoholic beverage of Romulan origin with a characteristic blue color (ranging from a pale sky blue to a dark midnight blue). It took a while to ferment and Doctor Leonard McCoy joked that he used it for "medicinal purposes." (Star Trek II: The Wrath of Khan)
In 2263 of the alternate reality, Montgomery Scott ordered three Romulan ales for himself at James T. Kirk's birthday party on Starbase Yorktown. (Star Trek Beyond) In the prime reality, Romulan ale was illegal in the Federation from at least the early 2280s to the late 2370s, but many Starfleet officers had actually sampled it at one point or another. A rare exception was Admiral William Ross. (Star Trek II: The Wrath of Khan; Star Trek VI: The Undiscovered Country; Star Trek Nemesis; DS9: "Inter Arma Enim Silent Leges")
Admiral James T. Kirk was given a bottle of Romulan ale dated 2283 as a birthday gift from Doctor Leonard McCoy in 2285. (Star Trek II: The Wrath of Khan) Captain James T. Kirk was able to procure some for his crew and served it at a state dinner with Klingon High Chancellor Gorkon in 2293. When Brigadier Kerla said he thought it was illegal, Kirk wryly noted it was "one of the advantages of being a thousand light years from Federation headquarters." However, the ensuing hangover convinced him to order the galley to no longer have the beverage served at diplomatic functions. (Star Trek VI: The Undiscovered Country)
In 2362, Fallit Kot hijacked a shipment of Romulan ale. He was sentenced to eight years in a Romulan labor camp. (DS9: "Melora") In 2366, the replicators aboard the USS Enterprise-D could not replicate Romulan ale; it would have required programming the devices with the molecular structure of the beverage. The recipe was not readily available, due to the lack of information and communication between the Federation and the Romulan Empire. (TNG: "The Defector") The Zibalian trader Kivas Fajo had twelve cases of Pratorean Romulan ale before the Federation confiscated them in 2366. These cases were mentioned in a listing of Fajo's collections in his Federation biographical data file. (TNG: "The Most Toys", okudagram)
Romulan ale was a challenge even for species as stout as the Klingons, who were usually quite resistant to the effects of alcohol. After drinking a sizable amount of Romulan ale in celebration of William T. Riker and Deanna Troi's wedding in 2379, Worf, obviously suffering its effects, stated, "Romulan ale should be illegal," to which Geordi La Forge replied, "It is." Captain Jean-Luc Picard and Shinzon later also shared a glass of the drink in a side room of the Romulan Senate chambers. (Star Trek Nemesis).
This was possible as Romulan ale had been legal in 2375 during the Dominion War, when the embargo was lifted as part of the alliance between the Federation and the Romulans. It was served aboard the USS Bellerophon on its way to Romulus and on the planet itself during a reception. Senator Cretak, Admiral Ross, and Doctor Bashir shared a glass of the drink in the mess hall of the USS Bellerophon. (DS9: "Inter Arma Enim Silent Leges")
Background information Edit
The identification of the blue drink Picard and Shinzon share is from the script of the film. It is also revealed there that Romulan Ale was served at Riker's bachelor party and that Worf is still hung-over from the ale he drank at that party.
Romulan ale has always appeared some tint of blue. Blue beverages were also seen in Romulan contexts in TOS: "The Enterprise Incident" and DS9: "In the Pale Moonlight". In the latter episode, the blue beverage is identified as kali-fal, which may be the Romulan name for "Romulan ale": they both appear to be bluish liquids that pack a wallop. However, kali-fal is known to have a potent aroma, whereas Romulan ale was never mentioned to have such an aroma.
Enterprise officers also enjoyed blue colored drinks in TOS: "Space Seed" and "Is There in Truth No Beauty?". However, the blue drink in the latter was explicitly identified as Antarean brandy on-screen.
According to Star Trek 25th Anniversary Special, the contents for the food props during the dinner scene aboard the Enterprise-A in Star Trek VI: The Undiscovered Country were: plastic sperm whale, hardboiled "Klingon" egg, unspecified flower species, chicken a-la-king, and blue Kool-Aid for the Romulan Ale.
Per the Star Trek Cookbook, the Romulan ale shown on the shows was actually Glacier Freeze Gatorade mixed with seltzer.
The Las Vegas Star Trek Experience sold bottles and six-packs of "Romulan ale", a blue lager-style beer made by Central American brewery Cerveceria La Constancia on exclusive license from Paramount Pictures. The Klingon actors were scripted to claim that "you Humans cannot hold your neurotoxins!"
In real life, the book 488 Bacardi Rum Based Cocktails included a recipe for "Real Romulan Ale", a mixture of Bacardi rum, Everclear alcohol, and Blue Curaçao liqueur. 
In Star Trek Beyond, after Montgomery Scott is heard in the background ordering three Romulan ales for himself, he is shown drinking a similarly brownish liquid. The Romulan ale bottle seen in Star Trek Nemesis was sold off on the It's A Wrap! sale and auction on eBay. 
Diane Duane's The Romulan Way states that Romulan ale is brewed from grains such as kheh, and states that the primary reason it is difficult to imbibe is that it is harsh to the throat. Terise Haleakala-LoBrutto, one of the protagonists, comments that the Romulans drink it in part to prove that they can. The book also has a form of wine that is easier on the throat but more alcoholic.
The Romulan tutorial missions in Star Trek Online indicate that the ale produced on Virinat, a Romulan colony, is brewed from honey-like secretions of khellid insects.