To lobby in American politics means to talk to legislators in private to convince them to vote a certain way on specific legislation. As a noun, a "lobby" is an organization that lobbies legislatures, either for pay from a sponsor or as a public service.
The term was in bad odor in the early 20th century, with many people (like Woodrow Wilson) convinced that it undermined representative government by skewing the thinking of legislators.
In reality, lobbyists spend most of their efforts providing information on specific legislation, pro or con, and letting the information sway the legislator. Bribery has always been illegal and hints of bribery destroy the effectiveness of a lobbyist.