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A Birthday cake
A birthday is the anniversary of the day of an individual's birth. Humans particularly enjoy celebrating birthdays, typically with a birthday party.
On September 2nd, 2151, Malcolm Reed celebrated a birthday. His parents told Captain Archer that they hadn't seen him on his birthday for "quite a few years". Archer ordered Hoshi Sato to find out what Malcolm's favorite food was so they could have a birthday meal. Hoshi later found out from Doctor Phlox that it was pineapple, and the crew surprised him with a pineapple birthday cake. (ENT: "Silent Enemy")
In 2154, Commander Charles Tucker III told T'Pol that she looked nice, "kind of like an old oil painting." T'Pol replied that she was not old, as she would only be 66 years old on her next birthday. (ENT: "Zero Hour")
On March 22nd, 2285, James T. Kirk reluctantly celebrated his 52nd birthday when his close friend Dr. Leonard McCoy gave him a pair of 18th century eyeglasses. He would later pawn these in 1986 San Francisco. When Spock asked, "Excuse me, weren't those a birthday present from Doctor McCoy?" Kirk replied, "And they will be again, that's the beauty of it." Spock gave him an antique copy of A Tale of Two Cities by Charles Dickens. (Star Trek II: The Wrath of Khan; Star Trek IV: The Voyage Home)
In 2333, Sarah Sisko abandoned her family just after Benjamin Sisko's first birthday. (DS9: "Image in the Sand")
Just before Julian Bashir's seventh birthday in 2348, he was taken to alien hospital on Adigeon Prime by his parents Richard and Amsha Bashir and began resequencing his DNA. (DS9: "Doctor Bashir, I Presume")
In 2356, young Annika Hansen celebrated her birthday on the USS Raven with her parents Magnus and Erin Hansen. (VOY: "The Raven")
In 2364, T'Shanik told Wesley Crusher that he did not look like he met the age requirements for the Starfleet Academy entrance exam. In response, he told her that he was turning sixteen the next month, to which Oliana Mirren announced "Happy birthday!" (TNG: "Coming of Age")
In 2367, William T. Riker celebrated his birthday in Ten Forward. (TNG: "Future Imperfect")
In 2368, Geordi La Forge told Data he could not remember what he did for his birthday when he turned seven, but could clearly remember when he first got his Circassian cat at the age of eight. (TNG: "Violations")
Marc Brooks died five months before his 38th birthday, which would have been around stardate 44356.9 in the year 2367. (TNG: "The Loss")
In 2369, Miles and Keiko O'Brien went back to Earth to attend the 100th birthday of Keiko's mother. (DS9: "Dax")
The same year Jadzia Dax suggested that it was Quark's birthday because of all the Ferengi vistors who arrived at the station. (DS9: "The Nagus")
In 2370, after Lieutenant Worf returned to the USS Enterprise-D from a Bat'leth competition from Forcas III, the crew threw him a surprise birthday party in his quarters, something he was not pleased about. (TNG: "Parallels")
In 2371, Gul Dukat's son Mekor celebrated his 11th birthday. Dukat had promised to make the time to take him to an amusement center in Lakarian City, but was unable to so, as he and Commander Benjamin Sisko were busy tracking down the stolen USS Defiant. (DS9: "Defiant")
Ensign Quintana's birthday in 2371 was around stardate 48423.2. Kira Nerys gave him a free hour in the holosuite as a birthday present. (DS9: "Meridian")
Later that year, Seska lamented to B'Elanna Torres that her brother's birthday was coming up and that she would not be able to celebrate it with him since she was trapped in the Delta Quadrant on the USS Voyager. (VOY: "Prime Factors")
Doctor Julian Bashir considered his 30th birthday in 2371 a major turning point in his life, although the Human lifespan by then was so long that he was still in the first part of his life. (DS9: "Distant Voices")
For Kes' second birthday in 2371, the USS Voyager crew threw a surprise party for her. They explained the event (Kes found it exhilarating) and instructed her on the Earth tradition of making a wish and blowing out candles. Neelix made a seven layer Jimbalian fudge cake with l'maki nut icing and punch. Tom Paris replicated a locket as a gift. (VOY: "Twisted")
When Rom and the employees of Quark's gathered to strike in 2372, Quark mentioned that if they were having a party for him, his birthday was last month. (DS9: "Bar Association")
Benjamin Sisko commented in 2373 upon the death of the latter that Jake Sisko and Enrique Muniz shared the same birthday. (DS9: "The Ship")
Keiko O'Brien prepared a surprise birthday party for her husband Miles in 2373, but during the party she actually was possessed by a pah-wraith. For the party she prepared q'parol, which was a big success. (DS9: "The Assignment")
In 2373, the crew of the USS Voyager threw Ensign Ahni Jetal a surprise birthday party in the mess hall. (VOY: "Latent Image")
In an alternate timeline created by the Krenim weapon ship, Commander Chakotay gave Captain Janeway a pocket watch for her birthday. (VOY: "Year of Hell")
In 2374, Morn planned on traveling to Luria from Terok Nor for his mother's birthday. However, he encoded a message in one of her presents from Major Kira to give to Benjamin Sisko on Starbase 375. (DS9: "Favor the Bold")
In 2376, Janeway had to search the databanks for hours on end, but eventually found Lieutenant Commander Tuvok's birthday, and replicated him a cake with a candle on it. He claimed that this was a Human tradition, and Vulcans do not "blow out candles." In the end, he blew it out, claiming that "it was a fire hazard." (VOY: "Fury")
Neelix once recounted a happy memory while he and Tuvok were alone in a Jefferies tube, when Voyager's crew held his birthday party in the mess hall. (VOY: "The Haunting of Deck Twelve")
In 2378, Ensign Harry Kim hoped to speak with his mother for the first time in seven years in time for her birthday, due to Operation Watson being successful in establishing contact with Voyager. In a draw, he got one of the lowest isolinear chips, meaning he would be able to speak to her in a couple of months. However, Tom Paris swapped Kim's chip with his, which meant he would be able to speak with her in a couple days. Paris asked Kim in return to sing Happy Birthday to her. (VOY: "Author, Author")
In an alternate timeline, the crew of Voyager held Kes' ninth birthday party in the ship's mess hall in 2378. (VOY: "Before and After")