Places to see in Rome: Campo de’ Fiori
Campo de' Fiori is one of Rome's most ambiguous spaces. This piazza in the center of Rome, close enough to both Piazza Venezia and Trastevere over the river, fulfills a variety of purposes for people in the city. Its history dates back to ancient times and its most prominent figure, a statue of the man killed for disagreeing with church canon, dominates the center of the square. During the day, Campo de' Fiori hosts a large food market with Italian products, fresh produce and flowers. At night, the many restaurants and bars surrounding the piazza fill up with tourists and locals. If you're visiting Rome, it's definitely a fun area to check out between trips to various monuments and long walks in the center. 
History
Campo de' Fiori translates literally into "field of flowers". In the middle ages, this square was actually a meadow, but later was transformed into a main commercial center of Rome. Shops were set up all around the area and the connecting streets all are named after various professions (hat-makers, vinegar producers, tailors, etc.)
Campo de' Fiori was also Rome's execution grounds where criminals and church dissenters were hanged or burned to death. In fact, there's a statue in the center that honors the famous figure of Giordano Bruno, a Dominican Friar and philosopher/scientist. Because he disagreed with Church teachings on geocentricity (all planets and the sun revolve around the Earth) and other major Christian concerns, he was tortured and burned at the stake in the middle of this square. The statue was later erected by the freemasons as a form of apology and remembrance of where the Church's past wrongdoings.
To this day, Campo continues to be a center where people gather together, albeit with a less grim atmosphere.
The Market
During the day (Monday-Saturday mornings), Campo de' Fiori acts as one of the few vegetable markets in the city center. While prices are higher than you might find at other markets further outside the city, it's a fun experience to walk through the stalls overflowing with beautiful, fresh produce and interesting Italian products.
You can find everything from oil, wine and spices to artichokes being shucked by hand and locals haggling with shopkeepers.
I've picked up small bits here for dinner later and a bouquet of flowers from one of the many flower stands near the fountain.
If you're looking for a quick snack, there are options for fresh squeezed orange juice and fruit cups, while at many of the bars surrounding the market you can sit down and have a coffee and snack.
Nightlife
If you're looking for an authentic Roman nightlife experience, Campo is not your place. This square is full of tourists (notoriously American) and study abroad students who drink to excess on overpriced, poor quality drinks. You might have luck grabbing a quick glass of wine or beer from one of the lower-key spots, but if you're interested in a nice night out with a less insane atmosphere, try out the neighborhoods of Monti, San Lorenzo and even parts of Trastevere.
On the other hand, if you're homesick and want to find other travelers, this square is a great place to meet people who speak English. Just make sure not to come here every night of your stay!
For more information on Rome's other markets, check out this list of great vintage markets in Rome here. For other shopping recommendations, I have a roundup of where to shop in Rome here.