The Roman Forum (a latin word meaning open space or market place) was the physical, governmental, and corporate center of the Roman world. The original Forum in Rome was designed by the architect Vitruvius.
Below is a map of Rome. In the center you can see where the Forum was located. The Capitaline Hill is the location for beginnings of Rome, Latium, and it extended out from there. in between this and the Palatine hill was a marsh that Vitruvius drained and used as his location for the Forum. It became the geographic center of the city.
The top center part of this reconstruction of Rome is the location of the forum. most of it is out of the picture, but you are able to see some of its splendor. Look northwest of the collesseum.
There are many buildings that typically surround a forum. Basilicas, shops, and temples, as well as triumphal arches and statues. The Rostra is one of the most famous places in the Roman Forum, because it is the public speaking platform that many famous speeches were given from. It was located at the head of the forum.
For an Illustration of the forum and an explanation of all the major buildings,
This is how the forum looks now.

The road going straight down the center of these pictures is the Sacra Via (the sacred way), it was the main road through the forum.
The forum. The Rostra is just to the right of the Arch of Septimius Severus (the left corner of the picture). On the right, the pillars you see are the remains of the Temple of Castor and Pollux, one of the Gemini twins.
A side view of the Rostra. As you can see, it was a large, long speaking platform where people could shout out their views and campaigns to passers by.
The Roman Forum has been a place of wonder and awe throughout the centuries. It is a popular tourist stop, and a wonderful subject for paintings.

Here is Panini's illustration of the Forum.
From the time Julius Caesar came to power, Emperors began expanding the Republican forum by building forums of their own nearby. The first Imperial Forum was built in 54 B.C. by Caesar. Most subsequent emperors built their own forums including Nerva, Vespacian, and Trajan, who built the largest which covered and area of 25 acres.
Here is an exquisite rendering of what Trajan's Forum might have looked like in its full use and splendour.
Trajan's Market- three stories of shops.