Palmer was born in King Street, South Shields, the son of George Palmer (1789-1866) a ship-owner. He attended Bruce's Academy in Percy Street, Newcastle. On leaving school, he entered his father's firm, but a year later, at the age of 17, he formed a partnership with Sir William Hutt, Nicholas Wood and John Bowes in the manufacture of coke.
At that time the northern coalfield faced severe problems in the London market because of difficulties in conveying coal by rail. Palmer and his brother George solved the problem by establishing a shipyard near Jarrow in 1851. The following year, the John Bowes was built, the first iron screw collier, with a capacity of 690 tons. The experiment was a complete success and the golden age of shipbuilding on the Tyne had begun.
Jarrow grew into a town of 40,000 and Palmer began to build warships. The Terror, a floating battery, was constructed and launched in three months for use in the Baltic during the Crimean War. Palmer further revolutionised the industry by substituting rolled armour-plate for forged plate. The first armour-plate mill was laid down at Jarrow for the manufacture of 'Palmer's rolled plates'. Warships built at Palmer's included the Lord Nelson (1908), the Hercules (1911), and the Queen Mary (1912) then the largest and most powerful battle-cruiser afloat. She was lost at the Battle of Jutland in 1916, with the loss of some 1200 lives. The Resolution (1916) saw service in World War II.
The outsider's point of view is expressed by Augustus Hare, the traveller and raconteur, who spent most of his time among the more picturesque areas of the North East. He visited his clergyman friend Edward Liddell in Jarrow in 1876 and found him:

'Amidst a teeming population of blackened, foul-mouthed drunken rogues, living in rows of dismal houses, in a country where every vestige of vegetation is killed by noxious chemical vapours, on the edge of a slimy marsh, with a distance of inky sky, and the furnesses vomiting forth volumes of blackened smoke. All nature seemed parched and writhing under the pollution...'

Meanwhile, Palmer introduced the cooperative principle for the benefit of his workmen and zealously promoted the welfare of Jarrow. He founded The Jarrow Mechanics Institute and the Palmer Memorial Hospital, and became the town's first mayor in 1875. When Jarrow was created a constituency, Palmer became its Liberal member till his death. His 1903 statue in Jarrow bears on the pedestal bronze plaques of Palmer's most famous ships, the S.S. John Bowes and H.M.S. Resolution.