A penny was a coin used on Earth in the 20th century as currency. It was worth one cent. In both Canada and the United States of America, a penny was worth 1/100th of a dollar. In Old Britain, a penny (better known in plural, pence) was worth 1/240th of a pound until 1971 and 1/100th of a pound thereafter.
When Gary Mitchell was increasing in power, Sulu stated that his ability was increasing geometrically and that it was "like having a penny, doubling it every day. In a month, you'll be a millionaire." Spock noted that soon Mitchell would "have attained powers we can't understand and can't cope with." (TOS: "Where No Man Has Gone Before")
When the USS Enterprise visited Sigma Iotia II, Kirk told Okmyx that he was tired of playing "patty cake with you penny-ante operators", pointing out that he was a small-time crook by using a poker reference. He then told him that the Federation was taking over, and if Oxymx played along, he would get a cut. (TOS: "A Piece of the Action")
While in the Sherlock Holmes holoprogram, Captain Picard picked up a two pence coin and referred to it as "tuppence — two pence", and kept it as a sign of good luck. Unfortunately, they were mugged moments later, with the ruffian demanding the coin they'd found, and any other money they had. (TNG: "Elementary, Dear Data")
Dr. Crusher used the old Earth adage "a penny for your thoughts" when talking with Picard and trying to get him to talk to her. When asked if she had one, she noted that she was sure that the replicator would have one on file. (TNG: "The Perfect Mate")
When planning to rob the casino in Vic Fontaine's old Las Vegas hotel in the holosuite program, Dr. Bashir quipped that "We have to see to it that Mister Zeemo never sees a penny of that money." (DS9: "Badda-Bing, Badda-Bang")
In the non-canon Star Trek: Voyager - Elite Force, during an in-game cinematic, Ensign Telsia Murphy said "a penny for your thoughts" to Munro. In response, he asked what a penny was. To this, Murphy replied, "I think it's an old coin. I think Ensign Paris might have one."