What is the meaning of the word Gasse in Austrian German; street or alley?
As far as I know there is no difference in the meaning of »Gasse« between German German and Austria German.
A Gasse is a small street in an urban environment. The length of a Gasse is just a few blocks (vienna's shortest Gasse is just 17 m long), and it has one or two lanes for driving cars, and normally one or two lanes for parking cars, and a sidewalk on each side.
But for each of this attributes you will find exceptions. The Hadikgasse in Vienna has 3 lanes, at some places even 5 (before crossings), and is many kilometers long. It is the beginning of B1 (Bundesstraße 1)*. But this is an exception. A normal Gasse is much smaller.
Other words often used in names of streets are:
(english: way): Like Gasse, but also used in rural environments. A famous exception is Rennweg in vienna, which is a long and big street.
Examples in Vienna: Ernst-Jandl-Weg, Rennweg, Rennbahnweg, Sileneweg
(english: alley): See above
Examples in Vienna: Falcogasse, Porzellangasse, Sensengasse, Silenegasse (Sileneweg and Silenegasse are at its closes point just 20 m away from each other)
(english: avenue): A street with trees on both sides. But there are also Alleen where there no longer are trees.
Examples in Vienna: Rotundenallee, Prater Hauptallee, Stadionallee
(english: street): Like Gasse, but normally longer and wider.
Examples in Vienna: Maria-Theresien-Straße, Schönbrunner Straße, Schönbrunner Schloßstraße
(english: promenade): A street, that was built for pedestrians.
Example in Vienna: Rudolf-Nurejew-Promenade
(englisch: walk): A Weg or Gasse, that was built for pedestrians.
Example in Vienna: Beethovengang
(english: ring): A street that surrounds the downtown (or parts of it).
One of the most important streets in Vienna is the famous Ringstraße, but you will not find a street with this name in Viennas streetmap. This street is devided into 9 sections, each of it has a different name, but the names of all nine sections end with -ring.
Examples in Vienna: Stubenring, Opernring, Universitätsring
(english: kay): A street that runs along a rivers bank.
Examples in Vienna: Handelskai (at river Danube), Franz-Joseph-Kai (at river Donaukanal)
Same as Kai: A street that runs along a rivers bank.
Examples in Vienna: Roßauer Lände, Erdberger Lände, Weißgerberlände (all three at river Donaukanal)
(english: line) Same as Kai and Lände: A street that runs along a rivers bank.
Examples in Vienna: Linke Wienzeile, rechte Wienzeile (at river Wien)
*(Bundesstraße is the 3rd category of streets in Austria; Category 1 is Autobahn, category 2 is Schnellstraße and Bundesstraße is category 3)
The question is most likely motivated by the fact that most street names in Austrian cities such as Salzburg and Vienna end in -gasse, not -straße. This is evidently related, in one direction or another (probably both), to the fact that it is more common in Austrian German than in German German to refer to a street in a city as a Gasse.
The word Straße is from Latin strata and originally referred to a paved road. The word Gasse is from a proto-Germanic word and originally referred to a road/street/path/passage that is laned by hedges or walls. A typical city street is paved and is laned by houses, so both original meanings make sense. In German German, Straße has become the standard term and Gasse is used only when its original sense is particularly strong, i.e. for narrow roads between houses or sometimes walls/hedges. In Austrian German, Gasse has become the standard term [added later:] regionally at least in Vienna. I am not sure what determines the alternative use of Straße in Austrian German - maybe Hubert Schölnast will shed some light on this.
PS: The results of web searches for "Straßenverzeichnis" in connection with München, Salzburg and Wien have convinced me that my original impression concerning Salzburg was wrong. Gasse, while being more common than in Munich is clearly not the default term there as it is rarer than Straße. However, Gasse has always felt like the default term to me in Vienna. The city is currently expanding west of the Danube, where I lived for a few years. Many of the streets there seem to get names ending in -gasse at a time when they are still surrounded mostly by fields and no dense building development is planned. On the other hand, I just checked, and it turns out that -straße is also used occasionally, for example in the new Seestadt quarter. I guess this is decided on a case by case basis, perhaps often determined primarily by who suggests a name and which part of the country they come from.
I (and I am Austrian) would prefer to translate Gasse as lane. I assume also lane having another meaning as alley.
See for example here.