(covers information from several alternate timelines)
Breasts was the collective term for physical protrusions of mammary glands on most species, both male and female. In most humanoid species, females were distinguished from males through larger breasts, a source of sexual attraction. Male breasts were typically useless, while females might use theirs for breast feeding. (TNG: "A Matter Of Honor"; VOY: "Deadlock")
Most cultures in the known galaxy required women to cover their breasts in public, and it was considered shameful to appear in public with them exposed. One exception to this rule was Betazoid wedding ceremonies, in which all attendees were naked. (TNG: "Cost of Living"; Star Trek Nemesis). If, for some reason, a woman ended up topless in public, they would frequently use an arm or any object to prevent their breast from being seen by others.
Such was the case when Enterprise NX-01 was captured and impounded by Cabal. The crew communicated through the EPS conduits to formulate a plan to escape. The only escape route was through the air vents and Ensign Hoshi Sato was small enough to fit. When she made her way to the outside of Lieutenant Malcolm Reed's room, her shirt was caught on a hook and she tried to get free, but ended up losing her shirt while sliding through. When she opened Reed's door, she held her hands over her breasts and told him to not say anything and to fetch her a shirt. (ENT: "Shockwave, Part II")
Until the reforms of Zek, Ferengi females were required by law to not wear clothing. (DS9: "Little Green Men") They also were prohibited from leaving their homes, due to the taboo of a Ferengi female appearing nude in public. In order to get around this, many Ferengi females would tightly wrap their breasts and pretend to be male in order to acquire profit and to travel away from Ferenginar. Such was the case of Pel in 2370, who in addition to wrapping her breasts, also enhanced her lobes in order to leave Ferenginar and work at Quark's. (DS9: "Rules of Acquisition") Additionally, in order to fill in for the ill Ishka, working on Zek's behalf, to prove the value of females to Nilva an influential FCA commissioner, Quark underwent cosmetic surgery and posed as a female Ferengi named Lumba. (DS9: "Profit and Lace")
Some dabo girls at Quark's wore dresses that showed part of their breasts so as to appear attractive to the male customers, who would look more at them than the game and thus lose money. (DS9: "The Abandoned")
"His eyes should flash,
and his breast protrude,
And this should be his customary attitude."
While on the Ba'ku planet, Deanna Troi asked Beverly Crusher if she had noticed how her boobs had started to firm up as a result of the metaphasic radiation that affected that planet, to which Crusher replied, "Not that we care about such things in this day and age," though Troi's snickering reaction suggested otherwise. Data, who overheard this conversation, repeated it to Worf. (Star Trek: Insurrection)
On the IKS Pagh, Klingons made fun of William T. Riker, who was posted on the ship to learn about Klingon culture. They told him that if he was afraid of the food, they could try to convince a female to breastfeed him. (TNG: "A Matter Of Honor")
The females on Draylax were known for having three breasts. Travis Mayweather once joked he knew it "first-hand, second-hand, third-hand". (ENT: "Broken Bow") The feline dancer on Nimbus III also had three breasts. (Star Trek V: The Final Frontier)