schweben
to float / hover
To move or remain in the air or in water, seemingly weightless and without visible effort or support.
Ein Heißluftballon schwebte langsam über die Stadt.
A hot-air balloon floated slowly over the city.

to float / hover
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to float / hover
To move or remain in the air or in water, seemingly weightless and without visible effort or support.
Ein Heißluftballon schwebte langsam über die Stadt.
A hot-air balloon floated slowly over the city.

Although 'schweben' (to float / hover) describes a kind of movement, the perfect tense is always formed with 'haben' (to have), not with 'sein' (to be).
Die Raumstation hat jahrelang im All geschwebt.
The space station floated in space for years.
The meaning of 'schweben' (to float / hover) depends on the subject. Physical things like a cloud or a feather literally float. Abstract things like a danger or a question hang in a figurative sense.
Eine einzelne Feder schwebte langsam zu Boden.
A single feather floated slowly to the ground.
In the abstract meaning of 'being uncertain', 'schweben' (to hang over) is often used with the preposition 'über' + dative to express what or who is affected.
Über dem Projekt schwebt die ständige Sorge der Finanzierung.
The constant concern about funding hangs over the project.
The idiom 'auf Wolke sieben schweben' (to be on cloud nine) is a fixed expression for great happiness or being in love.
Seit sie die Zusage für den Job hat, schwebt sie auf Wolke sieben.
Ever since she got the job offer, she has been on cloud nine.
to fall
'fallen' and 'schweben' both have to do with a position in the air or in water. The difference is the movement and the effect of gravity. With 'fallen', something moves downward, often quickly and uncontrollably. With 'schweben', something stays lightly in the air or in water, or moves there calmly. 'fallen' suggests weight and downward movement, while 'schweben' suggests lightness and calm.

Der Ballon schwebt über dem Garten.
The balloon floats above the garden.
Ein Blatt fällt vom Baum auf den nassen Weg.
A leaf falls from the tree onto the wet path.