brechen
to break
To split something hard or brittle into two or more pieces by force. This also includes bone injuries.
Er hat den Ast mit einem lauten Knacken gebrochen.
He broke the branch with a loud crack.

to break
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to break
To split something hard or brittle into two or more pieces by force. This also includes bone injuries.
Er hat den Ast mit einem lauten Knacken gebrochen.
He broke the branch with a loud crack.

'Brechen' (to break) can be used transitively (with an accusative object) or intransitively (without an object). Transitive means that someone or something actively breaks something. Intransitive means that something breaks by itself.
Sie bricht den Schokoladenriegel in zwei Hälften.
She breaks the chocolate bar into two halves.
When someone breaks a bone, German uses a dative construction: 'sich (Dativ) etwas (Akkusativ) brechen' (to break something). The person is in the dative, and the body part is in the accusative.
Ich habe mir beim Skifahren das Bein gebrochen.
I broke my leg while skiing.
In this meaning, 'brechen' (to break) is used with abstract nouns that represent a rule or agreement. Typical collocations are 'ein Versprechen brechen' (to break a promise), 'das Gesetz brechen' (to break the law), 'sein Wort brechen' (to break one's word), or 'das Schweigen brechen' (to break the silence).
Er hat sein Versprechen gebrochen, pünktlich zu sein.
He broke his promise to be on time.
When the subject of the sentence is 'Wellen' (waves), 'brechen' (to break) is always used intransitively. It describes the natural movement of waves as they reach the shore or obstacles.
Wir saßen am Strand und sahen zu, wie die Wellen am Ufer brachen.
We sat on the beach and watched the waves break on the shore.
In the meaning of 'sich übergeben' (to vomit), the verb is always used reflexively: 'sich brechen' (to vomit). This formulation is considered somewhat more formal or a euphemism (a softened expression).
Dem Jungen wurde auf der langen Autofahrt schlecht und er musste sich brechen.
The boy felt sick on the long car ride and had to vomit.
In a sports or competitive context, the most common and almost the only collocation is 'einen Rekord brechen' (to break a record).
Der Schwimmer hofft, bei den Olympischen Spielen den Weltrekord zu brechen.
The swimmer hopes to break the world record at the Olympic Games.
When 'brechen' (to refract) is used in a scientific context with 'Licht' (light) or 'Schall' (sound) as the object, it means 'to refract'. The subject is the medium that causes the refraction (e.g., water, glass).
Ein Prisma bricht das weiße Licht in seine Spektralfarben.
A prism refracts white light into its spectral colors.
to injure / hurt / violate
'verletzen' and 'brechen' can both refer to damage to the body, especially bones. 'Sich verletzen' is more general: you can have a wound, pain, or some other physical harm. 'Sich etwas brechen' is more specific and means that a bone is broken. There is also some overlap with rules: you can 'eine Regel verletzen' or 'eine Regel brechen' (to violate/break a rule). 'Eine Regel brechen' often sounds like deliberately not following it, while 'ein Recht oder eine Pflicht verletzen' (to violate a right or duty) sounds somewhat more formal. They can be confusing because both words occur with the body and with rules, but 'brechen' only fits when something is broken or when rules are not followed.

Lisa ist auf der Treppe gefallen und hat sich den Arm gebrochen.
Lisa fell on the stairs and broke her arm.
Beim Fußball hat sich Tom am Knie verletzt.
Tom injured his knee while playing soccer.