fächern
to fan
To move air with a fan or a similar object in order to cool yourself or someone else.
Sie fächerte sich mit der Speisekarte Luft zu, weil es so heiß war.
She fanned air toward herself with the menu because it was so hot.

to fan
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to fan
To move air with a fan or a similar object in order to cool yourself or someone else.
Sie fächerte sich mit der Speisekarte Luft zu, weil es so heiß war.
She fanned air toward herself with the menu because it was so hot.

The verb 'fächern' (to fan) is derived directly from the noun 'der Fächer' (the fan). The action of the verb imitates the movement or shape of a fan.
Weil ihr heiß war, nahm sie einen Fächer und begann, sich Luft zu fächern.
Because she was hot, she took a fan and began fanning air toward herself.
When you cool yourself or someone else, the construction 'jemandem (Dativ) Luft zufächern' (to fan air toward someone) is often used. The 'zu' is a separable verb particle here.
Kannst du mir bitte etwas Luft zufächern?
Could you please fan a little air toward me?
In the meaning 'to spread out', the verb is always used reflexively: 'sich fächern' (to fan out). The subject of the sentence is the thing that arranges itself in a fan shape.
Der Croupier fächerte die Karten auf dem Tisch.
The croupier fanned out the cards on the table.
to blow
'blasen' and 'fächern' can both mean that air is being moved. 'blasen' is more general: you push air out with your mouth or move air with a device. 'fächern' is used when you move air back and forth with a fan or a similar object, usually so that someone feels cooler. The words can be confused because the result can be similar: you feel air. The difference lies in the type of action: with 'blasen', the air comes directly from the mouth or a device; with 'fächern', it is created by moving an object.

Im heißen Bus fächert sie sich mit einer Zeitung Luft zu.
On the hot bus, she fans herself with a newspaper.
Du hast den Staub vom alten Buch geblasen.
You blew the dust off the old book.