Dragon Ball's Goten and Trunks have mixed blood. Their fathers, Goku and Vegeta, are full-blooded Saiyans. Their mothers, Chichi and Bulma, are full-blooded humans. This makes them half-Saiyan and half-human. The former half of their heritage has granted them a lot of perks, including immense strength and the ability to become Super Saiyans. However, there's one Saiyan trait that seems to elude these two-- their monkey-like tails.
Despite having half Saiyan, neither Goten nor Trunks have shown signs of growing or ever even having a Saiyan tail. Nor has Bulla, for that matter. This is strange as Gohan, Goku's other son, is also a Saiyan hybrid and he definitely had a tail. To explain this apparent anomaly, several explanations have been put forth by the fans and even through the Word of God.
The first solution is that Goten and Trunks had their tails cut off. This could have been done for a few reasons. The first explanation is that it would help them blend in better as Earthlings. It could also have been done so they wouldn't run the risk of turning into Great Apes at the sight of a full moon. The latter explanation in particular makes sense as it was also used for Gohan and, later, Broly. Whatever the case, losing the tail certainly seems like the optimal choice.
Another potential answer was actually provided by Akira Toriyama, the original writer of Dragon Ball. When asked about the matter in an interview, he said that Saiyan tails are a recessive trait. This would explain how Gohan can have a tail, but not Goten. It would also explain why other Saiyan hybrids like Pan and Bulla don't have tails, either. However, not all fans are satisfied by this answer, so they continue to look for more possibilities.
Many fans believe that Toriyama simply forgot about Saiyan tails over time. The author is known for having improvised much of the original Dragon Ball series. As tight as he was able to keep his narrative, there are definitely signs that he forgot things here and there. He's even gone on record to say he forgot all about Super Saiyan 2. It wouldn't be surprising to find out that Saiyan hybrid tails were among the things Toriyama lost track of.
Fueling the Toriyama-forgot theory is how long it had been since Toriyama needed to draw a Saiyan hybrid with a tail. The last time he had to draw them like this was when Gohan turned into a Great Ape and fought Vegeta. After this, the next hybrid that was put to paper was Future Trunks. If Toriyama had remembered that hybrids could have tails, he would have given one to Trunks at this point.
There's also the possibility that Saiyan tails in general were always meant to be phased out over time. Most of the living, canon Saiyans have either had their tails removed or, supposedly, never had them, to begin with. The only exceptions to this are Paragus and Tarble, and even they're not much to go on; one's dead and the other hasn't been seen on-screen since 2008. If tails were ever meant to be reintroduced, it would have been with the Universe 6 Saiyans, yet they are specifically stated to have evolved their tails away. All these missed opportunities make it difficult to imagine when or even if Saiyan tails will ever make a full-on return to the mainline Dragon Ball series.
As for Goten and Trunks, they likely won't be seen with tails any time soon, either. Whether theirs were removed or never existed, everyone seems content to have these two walking around without them. Plus, it saves artists and animators the trouble of drawing them.