Several characters in Dragon Ball have tails (しっぽ, Shippo) that, depending on the character, can be used in several different ways.
All Saiyan's 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. Full blooded Saiyans such as Raditz or Nappa wrapped their tails around their waist in a belt like fashion, likely to keep it from being gripped easily by an opponent.
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. 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.
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.
Goten's tail is shown in one of the anime openings to the show only. It appears that it was removed sometime before Gohan gives Goten the Flying Nimbus.
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.
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, Broly's clone has a tail as an adult before he is transformed into Bio-Broly.
Bardock is one of the only Saiyans (Other than the actual residents of Planet Vegeta) to keep his tail until his death. He usually wears it 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. It is incinerated along with the rest of his body by Frieza.
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.
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.
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 is about to kill Hirudegarn when Goku stops him, and kills the monster himself with Dragon Fist.