Favorite thing: Tripoli's old city, with its medieval streets, hammams, madrassas and mosques, is a fascinating place to wander around. Most of it dates from the Mameluke construction period in the 14th and 15th centuries.
The origin of the name Tripoli (Trablos today, in Arabic) comes Tripolis or the "Tri-city". Because of the rocky promontory upon which it's built, it looks like there are 3 cities instead of one. Today Tripoli is the largest city in the north of Lebanon, with a population of about 500000 people. It's 85 kilometers from Beirut.
Fondest memory: I liked the atmosphere of oderly chaos, and the friendliness of the people - especially Mahmoud, an elderly man we met who decided to show us around: he spoke only solid arabic, we spoke none of it! Still we understood each other easily, we talked and laughed, and ended our walk with an invitation to his home to meet his family and have coffee.
Tarablus (Tripoli) is a very confusing city, as the old part of town doesn't seem to follow any rules of town planning so it is easy to get absolutely lost. A nice place to get lost in though. The tourist office in Beirut has a map highlighting over 50 monuments to see, and away from the old city, is Al-Mina', the port area with a nice corniche and good street food.
Fondest memory: Watching the pigeons circling the houses at dusk. I saw this in Damascus as well - many old houses keep pigeons on their roofs and let them out at dusk. The idea of the 'game' is to disrupt the flight of other people's pigeons, trap them and breed from them. Some of the pigeons are worth loads.