Favorite thing: Inside the old walled city there is only one street that can be called so: the main street from Harar Gate to the main square. That will be your reference point if you get lost, the place to meet someone and so. The rest of the streets are really narrow alleys that make an impossible maze.
Harar old town is surrounded by old walls and has a few gates around. Those gates are not in use anymore, they are simply remains of the past. But Harar was once a well protected city in which outsiders were not welcome.
Nowadays there is not much left of the walls, and the remaning parts are not well preserved, so don't expect real amazing walls, just the memory of old times remains.
Favorite thing: The balconies in the old town reminded me those I saw in Zanzibar (Tanzania), with all that wooden embroderies and galleries. Most of them are poorly conserved and would need some restoration, but only those at Rimbaud House are somehow restored.
Walking about the maze of narrow alleys in Harar you can feel you are in any little town of Morocco, Tunis, etc. This is something you won't feel anywhere else in Ethiopia. The rest of the country has its own personality, so unique, neither "black Africa" nor "muslim Africa" but Harar is definetly a "muslim city".
Some parts of the town even reminded me places in Andalucia (south Spain) with all those white washed houses.
Harar has very typical houses, rather different from the rest of Ethiopia. Being a muslim city, they reminded me those I had seen in Morocco or Tunisia.
There is an "official" Typical House, at the Harari National Centre, near the Erer Gate. But if you hire a guide he will take you to more normal houses, where people live and you can see their daily life there. Basically, it's carpets on the floor and baskets on the walls.