beben
to quake
To make a violent, trembling movement. Often used for the earth during an earthquake or for the ground during a shock.
Bei dem Erdbeben hat die ganze Erde gebebt.
During the earthquake, the whole earth quaked.

to tremble / quake
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to quake
To make a violent, trembling movement. Often used for the earth during an earthquake or for the ground during a shock.
Bei dem Erdbeben hat die ganze Erde gebebt.
During the earthquake, the whole earth quaked.

The context shows what kind of 'beben' is meant. If it refers to 'die Erde' (the earth) or 'der Boden' (the ground), it is about a physical shaking. If it refers to a person, a body part, or the voice, it means a reaction to emotions or cold.
Als der schwere Zug vorbeifuhr, bebten die Wände des alten Hauses.
When the heavy train passed by, the walls of the old house trembled.
When a person trembles because of a strong emotion or cold, this is often expressed with the preposition 'vor' plus the feeling in the dative. Typical combinations are 'vor Wut beben' (to tremble with anger), 'vor Angst beben' (to tremble with fear), or 'vor Kälte beben' (to tremble with cold).
Die Kinder standen am Beckenrand und bebten vor Kälte.
The children stood at the edge of the pool and trembled with cold.
In the meaning of a shock or shaking, 'beben' is almost always used in connection with 'die Erde' (the earth) or 'der Boden' (the ground). It is the classic verb for describing earthquakes.
Bei der Explosion bebte der Boden in der gesamten Nachbarschaft.
During the explosion, the ground quaked throughout the entire neighborhood.
When a voice sounds unsteady, 'beben' is the appropriate verb. The reason is often given with 'vor' + dative or in a subordinate clause.
Ihre Stimme bebte vor Rührung, als sie die Dankesrede hielt.
Her voice trembled with emotion as she gave the thank-you speech.
to shake
'erschüttern' and 'beben' can both have to do with strong movement or trembling. The most important difference is the direction of the action: 'erschüttern' describes something causing something else to tremble. 'beben', on the other hand, describes the movement itself. The ground bebt, but an earthquake erschüttert the ground or a house. Learners may confuse the words because both often occur with earthquakes or strong impacts. Practically speaking: what moves bebt; what triggers this movement or strongly moves something erschüttert it.

Nach dem schweren Stoß bebte der Boden noch einige Sekunden.
After the heavy impact, the ground quaked for a few more seconds.
Ein lauter Knall erschütterte am Abend die Fenster des Hauses.
A loud bang shook the windows of the house in the evening.