der Bayer
Bavarian
A male person who comes from or lives in the German federal state of Bavaria.
Als echter Bayer trinkt er gerne Bier im Biergarten.
As a true Bavarian, he enjoys drinking beer in the beer garden.

Bavarian
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Bavarian
A male person who comes from or lives in the German federal state of Bavaria.
Als echter Bayer trinkt er gerne Bier im Biergarten.
As a true Bavarian, he enjoys drinking beer in the beer garden.

'Der Bayer' (the Bavarian) is the term for a male resident of the federal state of Bavaria. This type of word formation (region + ending) is typical for names of inhabitants in German.
Mein Nachbar ist gebürtiger Bayer.
My neighbor is Bavarian by birth.
The noun 'der Bayer' (the Bavarian) refers only to a person. To describe things that come from Bavaria, you must use the adjective 'bayerisch' (Bavarian).
Er ist ein Bayer und isst gerne bayerische Weißwurst.
He is a Bavarian and likes eating Bavarian white sausage.
In everyday language, the word 'Bayer' (Bavarian) is often associated with cultural clichés connected to Bavaria, such as lederhosen, beer gardens, and a particular way of life.
Er gilt als typischer Bayer, weil er so gesellig ist.
He is considered a typical Bavarian because he is so sociable.
The meaning 'Angehöriger des Stammes der Bajuwaren' (member of the Bavarian tribe) is used exclusively in historical or scholarly contexts. In normal usage, it always means the modern resident of Bavaria.
Das Museum zeigt die Siedlungsgeschichte der frühen Baiern.
The museum shows the settlement history of the early Bavarians.
Franconian / Frank
'Bayer' and 'Franke' both refer to people from German regions. A 'Bayer' is from Bavaria or lives there. A 'Franke' is from Franconia, a region that today lies partly in Bavaria but has its own regional identity. That is why it can be confusing: a Franconian may live politically in Bavaria, but often prefers to call himself a 'Franke' rather than a 'Bayer'. If you mean the federal state, 'Bayer' fits. If the region of Franconia is important, 'Franke' is more precise.

Mein Nachbar ist Bayer und kommt aus der Nähe von München.
My neighbor is Bavarian and comes from near Munich.
Herr Schwarz ist Franke und erzählt gern von seiner Heimatstadt Nürnberg.
Mr. Schwarz is Franconian and likes to talk about his hometown, Nuremberg.