All Saiyans have a monkey-like tail that allows them to transform into a Great Ape when they look at a full moon. When it is squeezed, it causes them agonizing pain, though Nappa and Vegeta become immune to this. Goku also became immune to this weakness, shown when Krillin attempted to squeeze his tail, to no avail. If a Saiyan's tail is cut off while they are in Great Ape form, they revert to normal.
Most of the Saiyans in the anime get their tails removed, but on some occasions, they grow back. The Daizenshuu description is that the tail will often suddenly grow back if the Saiyan is in danger; such as when it did during Goku's match against Giran or for Gohan during the battle against Vegeta. Full blooded Saiyans such as Bardock, Raditz, Vegeta (before it gets cut off) and Nappa wrapped their tails around their waist in a belt like fashion, likely to keep it from being gripped easily by an opponent.
Goku uses the Super Saiyan transformation while at the same time possessing a tail in Dragon Ball GT, but sometimes his tail does not change color to gold. The fur is still brown when Bardock turns Super Saiyan in Dragon Ball: Episode of Bardock. However, when Goku turns Super Saiyan 3, his tail fur always turns gold. When Goku and Vegeta turn Super Saiyan 4. their tail furs both take the color of their respective Super Saiyan 4 fur.
Goku loses his tail three times: the first time was when Puar transformed into a pair of scissors, and cuts it off after Goku transformed into a Great Ape in Pilaf's Castle. It ends up growing back during his fight with Giran at the 21st World Martial Arts Tournament. Later, Grandpa Gohan grabbed Goku's (still weak and untrained) tail during a fight at Fortuneteller Baba's Palace and accidently pulled it off. Then, his tail grows back in between the time after the defeat of Terror and Plague, and before the defeat of Sky Dragon by Goku. Lastly, Goku's tail was permanently removed by Kami at the end of Dragon Ball, but it was regrown by Old Kai in Dragon Ball GT to increase his power. Yet, the tail is not seen at the end of the GT series when Goku is an adult again.
After blowing up the moon to stop Gohan's rampage as a Great Ape, Piccolo yanked Gohan's tail off. In the anime only, the tail grew back, but was once again removed by Piccolo. It grew back for the second (third, counting the anime only instance of his regrowing his tail) and final time during the battle with Vegeta, and to stop his transformation, Vegeta cut off his tail with an energy disk, and it has not grown back since, though it has been featured in the movies Dragon Ball Z: The Tree of Might and Cooler's Revenge. In The Tree of Might it is cut off by Goku to end his transformation and, by doing so, protecting him from Turles' Kill Driver.
In the first episode of Dragon Ball Z, Raditz whips Krillin in the face with his tail. In Episode 3, Goku squeezes Raditz's tail, causing him terrible agony and getting him down for Piccolo to use another Special Beam Cannon, but Raditz tricks Goku into letting him go.
Vegeta's tail is cut off by Yajirobe and does not grow back. Though in Dragon Ball GT, it is regrown twice through artificial means. The first time is when Vegeta is under the control of Baby, and it is cut off by Goku. The second time is when it is regrown again so that he can become a Super Saiyan 4, but he does not retain this tail after he returns to his base form.
During the battle with the Saiyans, Piccolo devises a plan that involves him grabbing Nappa's tail. However, this plan fails, as Nappa is immune to this weakness and beats Piccolo, Gohan, and Krillin.
In the manga, Krillin asks Bulma about Trunks' tail and if she cut it off, but she never answered so it is unknown if Trunks ever had a tail.
Unlike his older brother Gohan, Goten was never shown with a tail in the series.
The movie villain Broly is shown to have a tail, as seen in flashback to his infancy. As an adult, his tail is not seen, but in a concept arts of Broly shows that his tail is hidden under his clothing. In the movie Dragon Ball Z: Bio-Broly, it is shown that Broly's clone has a tail as an adult before he is transformed into Bio-Broly.
Bardock usually wears his tail around his waist in a belt-like fashion. He uses it to transform into a Great Ape at the beginning of Dragon Ball Z: Bardock - The Father of Goku, and he still has it when sent back in time in Dragon Ball: Episode of Bardock.
Bardock's Crew had tails, and wore them in the same way Bardock did. They used them to transform into Great Apes along with Bardock at the beginning of Bardock - The Father Of Goku. Bardock's Crew are all killed by Dodoria and his elite soldiers.
Tarble seems to be the last living Saiyan to be seen still having his original tail. When he arrives on Earth to request Vegeta's help, his tail quickly gives away his Saiyan heritage. Unlike most Saiyans, Tarble does not wrap his tail around his waist in a belt like fashion.
The monster from the Muscle Tower, Buyon, can attack his opponents with his long and powerful tail. He uses this against Goku and Android 8. He also uses this attack as a boss in the video game Dragon Ball: Origins 2.
The Pirate Robot has a tail that he can electrify to attack and shock his opponent. The robot usually wraps its tail around its opponent before sending the surge of electricity through it. The robot uses this attack against Goku, after dragging him underwater. The Pirate Robot can also use this attack in the video games Dragon Ball: Advanced Adventure and Dragon Ball: Origins 2.
In battle, Frieza uses his tail mainly to slap enemies or strangle them. He does this to Vegeta and later Goku, though Goku bites him. When he is in his second form, it is cut off by Krillin with a Destructo Disk (causing him to threaten to eat Krillin) though he seemed to partially regrow his tail shortly before he transformed into his Third form, but when he turns into his fourth form, his tail is completely intact, though part of it is destroyed when he is hit with Goku's Spirit Bomb.
Cooler also sports a tail. Upon hearing of his brother's death while on a conquered planet, he is shown swatting away the remains of a throne behind him with his tail. During his fight with Goku, he attempts to whip his tail at Goku while fighting him underwater, before firing a Ki Blast at Goku.
Cell's tail is perhaps the deadliest of them all; he uses it to absorb people. He usually absorbs them by jabbing them with the stinger on his tail and then sucking them in, one bit at a time. This is a painful procedure that kills the victim being absorbed. When absorbing androids, however, the stinger on his tail opens up to suck the android in and swallow them whole. Cell does this to Android 17 and Android 18 in order to transform. In his perfect form, the tail retracts, though he does open it up again to spit out seven Cell Juniors. In Dragon Ball GT, he also extends it to absorb Goku in the same manner as 17 and 18, though he escapes.
In manga and anime, the "tail" theme seems to diminish after Cell achieves perfection.
Ozotto, the final boss of the arcade game Dragon Ball Z: V.R.V.S. can attack his opponent with his long and powerful tail.
In the movie Dragon Ball Z: Wrath of the Dragon, Hirudegarn's tail serves the same purpose as Cell's; he eats people with it, except he is much larger than Cell, and when his tail opens up, cords extend from it that wrap around victims and suck them in. After Tapion's Hero's Flute is broken, and Hirudegarn stomps on Hoi, Trunks cuts off Hirudegarn's tail with Tapion's Brave Sword. Trunks attempts to attack again Hirudegarn when Goku stops him, and kills the monster himself with Dragon Fist.