Old Cairo is a relatively small area but it is rich with history. Also known as Coptic Cairo, Fustat (in reference to the first Muslim city established there), and Masr al-Qadima to the locals, it has been inhabited since the 6th century BC. It has been a Roman fort protecting trade routes, a Christian city from around the 5th century AD, a Muslim army camp from 641 AD, then Egypt's capital city until yet another conquest in the 10th century.
The main interest these days is in its role as Coptic Cairo. The narrow cobbled streets contain the Religious Compound, full of churches including the Hanging Church (dedicated to the Virgin Mary and still in use), the oldest synagogue in Egypt, the remains of the Roman fortress, and the Coptic Museum. Just northeast is the site of ancient Fustat which contains the oldest mosque, Amr Ibn al-Aas. The rest of the area is interesting for the Zabaleen, people who live in a shanty town and sift through Cairo's huge amount of garbage to reuse and recycle it.
The easiest way to get to Old Cairo is by taxi, cheap if adventurous - the cars can be old and the driving is crazy! There is also a new metro system in Cairo. So far it is limited in area but signage is in English and the stop Mar Girgis will get you to Old Cairo.