Forum originated from the Latin word foris which meant outdoors, or outside (forest, forensic and foreign all originated from the same word). From the perspective of Bible History, a forum was a Roman (see Ancient Empires - Rome) public meeting or assembly area that took a number of different forms, ranging from a public market (equivalent to the Greek agora) to a sporting arena. Various government buildings, such as a basilica (a basilica was originally a Roman public building, but since the Church of Rome was created in the image of the Roman empire, it adopted many of the pagan Roman terms, such as basilica) and pagan temples were usually near the forum.
The Appii Forum
The only forum mentioned as such in the Scriptures is the Appii forum, also known as the forum of Appius, or the market of Appius. After departing from Malta, it was on the apostle Paul's (see Paul's Ministry) route as he approached Rome. There, Paul was met by some members of the true church at Rome at the time before the majority of its people became apostate (see the Fact Finder question below).
"And after three months we departed in a ship of Alexandria, which had wintered in the isle, whose sign was Castor and Pollux."
"And landing at Syracuse, we tarried there three days."
"And from thence we fetched a compass, and came to Rhegium: and after one day the south wind blew, and we came the next day to Puteoli: Where we found brethren, and were desired to tarry with them seven days: and so we went toward Rome."
"And from thence, when the brethren heard of us, they came to meet us as far as Appii forum, and The three taverns: whom when Paul saw, he thanked God, and took courage."
"And when we came to Rome, the centurion delivered the prisoners to the captain of the guard: but Paul was suffered to dwell by himself with a soldier that kept him." (Acts 28:11-16 KJV)
Fact Finder: How did many of the true Christian church at Rome become apostate? Who led them astray?
See Simon Magus