I realized that when I think of the advent and spread of electricity, I always think about it in relation to the light bulb and lighting homes, but that is far from the only use of electricity today. Had there been other practical purposes before the light bulb or was that the first?
The telegraph was popularized and spread around the world long before the electric light became common. The spread of the telegraph allowed electrical experimentation on a large scale ─ often literally as people coped with the behavior of electricity over long wires.
The 1700s were the great start of electrical experimentation. Benjamin Franklin tamed wild electricity with the lightning rod in 1749, and at the end of the century Allessandro Volta produced domesticated electricity with the Voltaic pile. This made electricity above the level of static available.
By 1801, the Royal Institution had the most powerful Voltaic pile in the world. Their new professor, Humphry Davy, delivered lectures on galvanism and ran enough power through a strip of platinum to make it incandescent. By 1806, he invented the arc lamp. In 1807, he founded electrochemistry, using it to produce pure elemental potassium and sodium.
Better use of electricity waited on developing something better than batteries. The first electrical generator, turning motion into electricity, was invented by Sir Humphry's successor at the Institution, Michael Faraday.
It was by his research on the magnetic field around a conductor carrying a direct current that Faraday established the basis for the concept of the electromagnetic field in physics. Faraday also established that magnetism could affect rays of light and that there was an underlying relationship between the two phenomena. He similarly discovered the principle of electromagnetic induction, diamagnetism, and the laws of electrolysis. His inventions of electromagnetic rotary devices formed the foundation of electric motor technology, and it was largely due to his efforts that electricity became practical for use in technology.
That would be about 1832. Telegraphic systems - there were several at first - start up around 1835 and proliferate by the 1840s. I think /u/The_Alaskan is quite correct that this is the first public, practical, outside the lab use of electricity. Edison bulbs date to 1878 and are the basis for all radio and electronics.