Architecture of Salzburg
Obviously, Salzburg is a Baroque city and people with a specialist interest in architecture should focus on this period in the Altstadt. Note the early Baroque leisure palace of Hellbrunn with its trick fountains; the Salzburger Dom, a cathedral; some late-Baroque buildings such as the Kollegienkirche (in our opinion, Johann Bernhard Fischer von Erlach′s masterpiece); and the gardens of Mirabell Palace (the Mirabell palace itself was altered until the 19th century and is not particularly interesting in the architectural sense of the word).
Note also the city walls, most of them date to the first half of the 17th century - as well as a fair chunk of the Festung Hohensalzburg. The palaces of Leopoldskron and Klessheim are late-Baroque and demonstrate how the geometric outlay of the city was extended further to the outskirts the bigger it grew. For a nice church of pilgrimage, note the Basilica of Maria Plain. The Kapuzinerkloster (Capuchin Monastery) is a rather humble building, typical for the order.
Salzburg Architecture beyond Baroque
Other periods that were important for Salzburg basically reflect the economic history of the city: Boom periods such as the late Middle Ages or the late 19th century have left their marks, periods of recession such as the early 19th century rarely have left anything. Note the many Gründerzeit houses along the Salzach River and the many schools and public administrative buildings that were designed in Historicist style, generally after 1860, especially after 1880.
That was when "developmental aid" from Vienna and the rise of tourism finally allowed Salzburg to recover from the shock of loosing both its independence and most lucrative mines (salt and various metal ore deposits which were not sustainable anymore). A prominent architect of this period was Valentin Ceconi. For examples of his work, note the many representative villas along the Salzach near the city centre, but also the neighbourhoods of Schallmoos, Andräviertel and Froschheim.
With the economic boom after WWII ("Wirtschaftswunder"), the city grew rapidly and you will find a great deal of modern constructions - fortunately, most of them are located on the outskirts of the Altstadt. Few of them interfere with the Baroque and Medieval ensemble of Old Town houses. Note that in the sightseeing section, we have pointed out several modern buildings of Salzburg as examples for contemporary architecture.
The Wikipedia article on Salzburg contains some info on architecture
Salzburg Province & City on architecture