das Kästchen
small box
A small, often lockable container or small box used for storing objects.
Sie bewahrt ihren alten Schmuck in einem hölzernen Kästchen auf.
She keeps her old jewelry in a small wooden box.

small box / checkbox / square
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small box
A small, often lockable container or small box used for storing objects.
Sie bewahrt ihren alten Schmuck in einem hölzernen Kästchen auf.
She keeps her old jewelry in a small wooden box.

'Kästchen' (small box / checkbox / square) is the diminutive form of 'der Kasten' (box/crate/container). The ending '-chen' makes the word neuter ('das') and means 'small box'.
Er hat einen großen Kasten für Werkzeug und ein kleines Kästchen für Schrauben.
He has a large box for tools and a small box for screws.
In the meaning 'checkbox', 'Kästchen' (checkbox) is very often used with the verbs 'ankreuzen' (to check/tick) or 'auswählen' (to select). A common phrase is also 'ein Häkchen in das Kästchen setzen' (to put a check mark in the checkbox).
Bitte kreuzen Sie alle zutreffenden Kästchen an.
Please check all applicable boxes.
When referring to a small container, 'Kästchen' (small box) is often combined with verbs such as 'aufbewahren' (to keep/store), 'verstauen' (to stow away), or 'öffnen' (to open). Adjectives such as 'hölzern' (wooden), 'alt' (old), or 'geheimnisvoll' (mysterious) are also typical.
In dem alten Kästchen auf dem Dachboden fanden wir Liebesbriefe von unserem Großvater.
In the old small box in the attic, we found love letters from our grandfather.
The meaning 'field in a grid' can be recognized from the context of games, patterns, or paper. Typical formulations are 'ein Kästchen ausmalen' (to color in a square) or 'etwas in ein Kästchen schreiben' (to write something in a square/box).
Die Kinder malten die Kästchen auf dem karierten Papier in verschiedenen Farben aus.
The children colored the squares on the graph paper in different colors.
field
'das Feld' and 'das Kästchen' can both mean a small, clearly separated area on paper or on a screen. 'das Feld' is more neutral and is often used for input fields in forms, tables, or databases. 'das Kästchen' sounds smaller and often means a square that you check or fill in. It can also be a small square in a grid. The words can be confused especially in forms. For a text field or data field, you are more likely to say 'Feld'; for a small checkbox or a square, you are more likely to say 'Kästchen'.

Luis setzt ein Kreuz in das Kästchen neben seiner Antwort.
Luis puts a cross in the box next to his answer.
Im Feld für die E-Mail-Adresse fehlt noch ein Punkt.
There is still a dot missing in the field for the email address.