Grove was born in Swansea, Wales in 1811. He was a professor of physics in the London Institution from 1841 to 1846. He also served as a barrister and a judge of the Court of Common Pleas and of the High Court of Justice. Grove was the first to show that electrolysis, with a high-tension current, can take place through thin glass. The first experimental proof of dissociation was given by Grove, who showed that steam in contact with a strongly heated platinum wire decomposed into hydrogen and oxygen.

His nitric acid cell was the favorite battery of the early American telegraph (1840-1860), because it offered strong current output. By the time of the American Civil War, Grove's battery was replaced by the Daniell battery. As telegraph traffic increased, it was found that the Grove cell discharged poisonous nitric dioxide gas. Large telegraph offices were filled with gas from rows of hissing Grove batteries. As telegraphs became more complex, the need for constant voltage became critical and the Grove device was necessarily limited (as the cell discharged, nitric acid was depleted and voltage was reduced).

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