A bribe is a gift (often in the form of money) intended to gain favor with a person of influence. Most cultures consider bribery morally reprehensible, although in many cases (e.g. the Cardassian Union) it is widespread despite being frowned upon. The Ferengi Alliance is a notable exception; bribes are considered a normal part of day-to-day life, to the point that it is sometimes considered rude not to bribe someone.
The Cardassian military was rife with corruption. During the Occupation of Bajor, Quark was able to conduct illegal business at his bar, as he knew Gul Dukat could be bribed into looking the other way so long as it was nothing that would gain the attention of his superiors. Other officers like Boheeka were willing to do Quark a favor now and again if the price was right. (DS9: "Necessary Evil", "The Wire")
Officials in the Ferengi Commerce Authority (FCA) routinely bribe their way to power. Liquidators make up for this by charging citizens for virtually everything, even knowledge of what crime one has been charged with. The line between bribery and standard business proceedings often becomes blurred, such as at the Tower of Commerce, where visitors must pay three strips of latinum for the use of a chair while they wait and one strip if they wish to stand. (DS9: "Family Business")
When Rom had second thoughts about marrying Leeta, he considered bribing her into signing the standard Ferengi waiver females must sign prior to a wedding; however, he failed to take into account that the waiver would bequeath all of her property to him, making the bribe rather pointless. (DS9: "Ferengi Love Songs")
Part of Grand Nagus Zek's reforms, once he wrested control back from Brunt, was to make all bribes tax-deductible. Quark was outraged to hear this news, because he had never thought he would hear the "T-word" on Ferenginar. (DS9: "The Dogs of War")
Starfleet officers are often considered immune to bribes due to their moral code, which strictly prohibits such illicit behavior. However, they have been known to succumb to greed on occasion, becoming involved with organizations such as the Orion Syndicate. (DS9: "Honor Among Thieves")