In the episode "The One With A Chick And A Duck", Joey bought the chick from an animal sanctuary as a gift for Chandler after seeing (and misunderstanding) a news report about people who buy chicks and then find they cannot properly care for them.
Phoebe encouraged Chandler to return the chick but Chandler, discovering that the chick was in danger of being euthanized, did not have the heart and returned home with the chick and a duck. It turned out that, after hitting a few bumps (such as getting the chick stuck in the VCR) the two were perfectly capable of caring for the birds, and they were well loved.
A recurring gag throughout the series has Chandler and Joey treating the chick and the duck as their own children, a continuation of the earlier gag that Joey and Chandler acted like a married couple. At one point, Chandler punishes the duck for trying to eat the chick by sending him out to the hall; stating, "You stay out here and think about what you did!", greatly surprising Ross and Rachel. On another occasion, when Joey threatened to move out after Chandler kissed his girlfriend, Kathy, Chandler attempted to convince him to stay by arguing that the chick and the duck had been having a hard year recently, due to the apartment being robbed.
Eventually, when the chick unexpectedly grew into a rooster (Chandler and Joey believing that the chick was female prior to this development), Monica and Rachel made a bet with Joey and Chandler where they would give away the rooster and the duck if they lost, but Chandler and Joey would get the girls' apartment if they won; the transition was only temporary. On a later occasion, the duck swallowed Ross's engagement ring during the bachelor party for his wedding to Emily, requiring the duck to be taken to the vet to get it out. When Chandler moved to Monica's apartment, it is revealed that Monica hates them but disguises it as an allergy which prevents them from visiting.
After about Season 6 the animals disappeared (the duck was mentioned but not seen in Season 7 as having thrown up on Joey's bed), and in the final episode it was revealed that they had died of old age some time ago, but Chandler and the rest of the group had lied to Joey to save his feelings, telling him that they had gone to a special farm where they were very happy but could not be visited.
Chandler named the chick "Yasmine", after Yasmine Bleeth, an actress on Baywatch, a television program he and Joey frequently watched, though the chick later turns out to be a male when he grows into a rooster. In "The One With Ross's Thing", Joey and Chandler are discussing if the chick and the duck had a baby, Joey says "we could call it Chuck!" (ch(ick)(d)uck), Chandler responds with "or, Dick" "(d(uck)(ch)ick). Later in that episode, Chandler refers to the duck as Dick.
Chick Jr. & Duck Jr.
In the final episode, Joey bought Chandler a new chick and duckling as a housewarming gift (Joey named them "Chick Jr." and "Duck Jr." respectively).
In the end, Chandler tells Joey to keep them himself and, again after a few bumps (this time they got stuck in the foosball table), he even considered getting a goose, but Chandler quickly discouraged him, telling him he was "set with the poultry". The birds did not make an appearance on the Friends spinoff Joey, and their final fate is unknown (it is possible that they did in fact move in with Chandler and Monica).