Bribery is akin to what is commonly considered graft, payoffs, or influence peddling. Bribery involves public officials, government officials, witnesses — anyone acting for, on behalf of, or under the authority of the government. It can also describe a sporting official who “fixes an event”. Whether a Senator, a DOD subcontractor, a city councilmember, a federal witness, or a juror, it is considered a bribe whenever such an "official” trades influence in his or her official capacity for anything of value.
A bribe can be direct or indirect, it can include the giving of something of value or merely the promise to do so. In most situations, both the person offering the bribe and the person taking it can be charged with bribery. States as well as the feds have harsh penalties on those giving or receiving bribes.
US Code Title 18, Chapter 11, specifically §201-209, deals with the myriad details, definitions, and offenses that constitute bribery. They specifically deal with influencing official acts, influencing collusion to allow fraud, influencing the act or omission of an official duty, and influencing testimony. Laws against influencing jurors and officers of the court (US Code Title 18, §1504-1506) through bribery are distinct from influencing them through threat, force or writing, as well as obstruction of civil anti-trust investigation.
Characteristically the FBI rules over most types of bribery cases, from the bribery of public witnesses and officials to bribery in sporting contests to bribery concerning programs that receive federal funds. Bribery is just one of the specified illegal activities for which the US Postal Service has its own investigative jurisdiction when investigating certain types of money laundering (§1956) and monetary transactions in property derived from “specified unlawful activity” (§1957).
In addition to criminal penalties that include one to 15 years in prison and/or steep fines, the Attorney General may bring a civil action against any person who engages in conduct constituting an offense. The Attorney General may also petition the appropriate United States district court to bring civil action. Civil penalties can be as much as $50,000 per offense. There are similar penalties for the official who has received the bribe, as outlined in § 201.