St Peter's Square and Basilica - Vatican City in Rome
Whether you are religious or not, I can definitely recommend you checking out the Vatican, if you are in Rome. The historical importance of this city (country?) within the city of Rome is incredibly high, and you can find an overwhelming amount of art here, since the church never had any trouble affording the best artists to contribute to their impressive aesthetics and symbolism.
If you look closely in the picture above, you can see a line of people queueing to the left. The wait itself is quite a spectacle, a mix of tourists of all kinds, once-a-lifetime pilgrims, school classes and art lovers. The entrance is free, but there is a security check like you know it from airports, hence the wait.
pictures and words can never give this incredible place justice, so I won't even try to explain all of the things to see here. If you are interested in arts, or Christian history, or history in general, I would take a day (and a good guide) to get an overview of all the things offered here.
In the middle of the basilica, near the altar, there is a small stairway leading downstairs to the tombs, where all popes of the last century have been buried. It's a sacred place and lots of people come here for prayers. all in all there are 100 tombs inside and beneath the basilica...
Going up is a good idea, too, so you can pay a fee of 5€ to ascend to the dome on your own (there is an elevator if you want to save your steps for later).
On the walk down you pass the Vatican post office, which has a cult status since it has it's own stamps, so you can write postcards to your friends here and post them in the bright yellow post box.
You also get a great view over St Peter's Square:
There is an endless amount of things things to know and say about this place. Part of it is the uncountably high number of mysteries, scandals and a portion of magic that comes with it. I can definitely recommend going and seeing all the precious things with your own eyes.