As u/otblock57 already mentioned, you will find many dialects in Austria that stem from the Bavarian and Alemannic branches of the German language. However, this is a very technical distinction as, to many people, the "typical" Bavarian (mainly spoken in the 'Freistaat Baiern', a region in south-east-Germany) sounds rather different from, say, the common dialects in Steiermark (Styria) or Niederösterreich (lower austria). Alemannic variants are mostly prevalent in the regions of Vorarlberg and some regions in Tyrol.

All of this is describing most of the dialects here in Austria, while our official language is simply German - our schools and authorities use "österreichisches Deutsch" (Austrian German) which is basically a localized variant of standard German. There are some grammatical and vocabulary differences, but speakers from most parts of Germany and Austria will generally understand each other very well.

EDIT: to provide an actual answer as to the national languages of both countries:

• Switzerland has French, Italian, German and Romansh as national languages- The majority of people there grow up with German as their primary language and they often learn one, sometimes two of the other languages. People fluent in all the national languages are rather rare.
• Austria has German as its national (and official language), while it also recognizes several minority languages, such as Hungarian and Slovenian. The vast majority of people here grow up with German as their primary language. Even those growing up with one of the minority languages as their primary will learn German in school and be somewhat fluent in it as either a second language, or even on the same level as Hungarian, Slovenian etc.