Turin was founded by the Romans more than 2000 years ago, but they didn't left much (the most famous roman building in Turin is probably the Porta Palatina). The main aspect that the roman left to the city is the city plan, in which all the streets are nearly always perpendicular one to the others.
The city started to become important when the dukes of Savoy moved here, so from the XVI century onward. A big part of the historical center was influenced by them, and it is characterized by a baroque style. They also built a lot of arcades to cover sidewalks of the main streets.
Another period in which a lot of new structure were built is the end of the XIX century, when the Savoy family became the Royal family of Italy (for a while Turin was also the Capital, but later it was moved to Florence and then to Rome). For example the Mole Antonelliana, which is the main monument of the city, was built in that period. The most of the "old looking houses" in the city are from this period.
In 1899 there was also the foundation of the FIAT, that modified the city (in particular the south part) with its plants and the workers houses.
Finally, after the WWII really a lot of people from the rest of Italy started to move in Turin to work in its industries, and in few years the city nearly doubled its inhabitants. So a lot of more recent apartment blocs were built, and the city enlarged a lot.
Some pictures:
-a general view
http://grandhotelsitea.it/wp-content...a_scoprire.jpg
From the city center:
-Piazza San Carlo (one of the main squares of the city center)
https://it.wikipedia.org/wiki/File:T..._San_Carlo.jpg
-some arcades
http://www.quartieri.net/immagini/fo...maPortici1.jpg
- Piazza Castello, with the "Palazzo Reale" (Royal Palace)
http://www.museotorino.it/images/71/...1.jpg?VSCL=100
-Piazza Vittorio Veneto, and the top of the Mole Antonelliana (the three colored band where put on the Mole to celebrate the 150th anniversary from the unification of Italy)
https://c1.staticflickr.com/7/6100/6...f5e76731_b.jpg
Some more recent building:
-the "lingotto", one of the firsts FIAT factories. Now it hosts a mall, a congress center an a lot of other stuffs
https://it.wikipedia.org/wiki/File:Lingotto-2.jpg
- and then there are a lot of apartment blocks, but those look the same everywhere.