der Hut
hat
A head covering that typically consists of a crown and a brim.
An sonnigen Tagen trägt er immer einen großen Strohhut.
On sunny days, he always wears a large straw hat.

hat
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hat
A head covering that typically consists of a crown and a brim.
An sonnigen Tagen trägt er immer einen großen Strohhut.
On sunny days, he always wears a large straw hat.

'Der Hut' means 'hat' in English. The English word 'hut' (a small, simple cabin) is 'die Hütte' in German. This is a common mix-up.
Bei diesem kalten Wetter brauchst du einen Hut und Handschuhe.
In this cold weather, you need a hat and gloves.
The word 'Hut' (hat) is related to the verbs 'hüten' and 'behüten' (to protect, to guard). This connection is still easy to recognize in the idiom 'auf der Hut sein' (to be on guard).
Ein Hirte hütet seine Schafe.
A shepherd tends his sheep.
In connection with the head covering, typical verbs such as 'einen Hut tragen' (to wear a hat), 'einen Hut aufsetzen' (to put on a hat), and 'einen Hut abnehmen' (to take off a hat) are used.
Er setzte seinen Hut auf und verließ das Haus.
He put on his hat and left the house.
The expression 'auf der Hut sein' (to be on guard) is often expanded with the preposition 'vor' and the dative to name the danger one should watch out for.
In dieser Stadt musst du vor Taschendieben auf der Hut sein.
In this city, you have to be on guard against pickpockets.
In the expression 'unter einen Hut bringen' (to juggle/reconcile), the things that need to be reconciled are the accusative object of the sentence.
Sie versucht, ihre vielen Termine unter einen Hut zu bringen.
She is trying to juggle her many appointments.
The idiom 'den Hut nehmen' means to give up a position or be dismissed. It is usually used with a possessive pronoun, e.g. 'er musste seinen Hut nehmen' (he had to step down).
Nach der Wahlniederlage musste der Parteivorsitzende seinen Hut nehmen.
After the election defeat, the party chairperson had to step down.
When showing respect for someone, you use the preposition 'vor' with the dative: 'vor jemandem den Hut ziehen' (to take one's hat off to someone).
Vor seiner Leistung kann man wirklich nur den Hut ziehen.
You really can only take your hat off to his achievement.
The idiom 'etwas auf dem Hut haben' (to be responsible for something) is colloquial. It means to be responsible for a task or area.
Wer hat heute den Küchendienst auf dem Hut?
Who is responsible for kitchen duty today?
helmet
'der Hut' and 'der Helm' are both things you wear on your head. The most important difference is their purpose. A hat is usually clothing or fashion and may protect you from the sun or rain. A helmet is hard and is meant to protect the head from injuries, for example when cycling, riding a motorcycle, or on a construction site. Learners may confuse the words because both are head coverings. When safety is important, you say 'Helm'; for normal clothing, you are more likely to say 'Hut'.

Wegen der starken Sonne trägt Herr Hofmann heute einen Hut.
Because of the strong sun, Mr. Hofmann is wearing a hat today.
Nele setzt vor der Fahrradtour ihren Helm auf.
Nele puts on her helmet before the bike ride.