die Form
shape
The outer shape, structure, or outline of an object.
Die Vase hat eine sehr ungewöhnliche Form.
The vase has a very unusual shape.

form / shape
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shape
The outer shape, structure, or outline of an object.
Die Vase hat eine sehr ungewöhnliche Form.
The vase has a very unusual shape.

'Die Form' usually means 'shape' or 'condition'. A document that you have to fill out is called 'das Formular' in German, not 'die Form'.
Füllen Sie bitte dieses Formular aus.
Please fill out this form.
The word family around 'Form' is very useful. It includes the verb 'formen' (to shape), the adjective 'förmlich' (formal), and the noun 'die Uniform' (uniform clothing).
Ein Töpfer formt aus Ton eine Vase.
A potter shapes a vase out of clay.
In the context of cooking, baking, or also industry, 'die Form' means a container that gives a mass a shape. Typical examples are 'die Kuchenform' (cake pan) or 'die Gussform' (casting mold).
Hast du eine runde Backform für mich?
Do you have a round baking pan for me?
There are many fixed expressions for talking about physical condition: 'in Form sein' (to be in shape), 'in Form kommen' (to get in shape), or 'außer Form sein' (to be out of shape).
Nach seiner Krankheit ist er völlig außer Form.
After his illness, he is completely out of shape.
In social situations, 'Form' refers to proper behavior. An important expression is 'die Form wahren', which means to behave politely and correctly, even when you are annoyed.
Obwohl er wütend war, wahrte er nach außen hin die Form.
Although he was angry, he outwardly maintained proper form.
The fixed expression 'in Form von' (in the form of) is used to describe the way or appearance of something. It is combined with a noun in the dative.
Die Hilfe kam in Form einer großzügigen Spende.
The help came in the form of a generous donation.
kind / way
'die Art' and 'die Form' can both describe how something is or how something appears. 'die Art' tends to emphasize the way, kind, or type. 'die Form' often emphasizes the outer shape, for example round, angular, or flat. But it can also mean the form of presentation, for example with a letter or an application. Learners may confuse the words because in some sentences you can hear both. If you are asking 'how is it done?' or 'what kind?', 'Art' often fits. If you are asking about the shape or external design, 'Form' is usually a better fit.

Vor dem Termin füllt Anna das Formular in schriftlicher Form aus.
Before the appointment, Anna fills out the form in writing.
Diese Art von Handy ist im Sommer besonders praktisch.
This kind of mobile phone is especially practical in summer.