Police were called by the ambulance service shortly before 01:40 BST after the discovery at Waterglade Industrial Park in Eastern Avenue, Grays.
The lorry driver, a 25-year-old man from Northern Ireland, has been arrested on suspicion of murder.
Essex Police said the lorry travelled from Bulgaria and entered the country via Holyhead, Anglesey, on Saturday.
Early indications are that 38 adults and one teenager have died, the force said.
Ch Supt Andrew Mariner said officers were trying to identify victims but anticipated a “lengthy process”.
“We have arrested the lorry driver in connection with the incident, who remains in police custody as our inquiries continue,” he said.
Prime Minister Boris Johnson said he was “appalled by this tragic incident”
He said: “I am receiving regular updates and the Home Office will work closely with Essex Police as we establish exactly what has happened. My thoughts are with all those who lost their lives and their loved ones.”
Home Secretary Priti Patel said she was “shocked and saddened by this utterly tragic incident”.
Thurrock MP Jackie Doyle-Price said it was “sickening news”.
“People trafficking is a vile and dangerous business,” she tweeted, adding that she hoped Essex Police could “bring these murderers to justice”.
Glen Freeland, from GSF Car Parts, which is close to where the bodies were found, said staff could not access their premises on Eastern Avenue.
“The manager went to get into work this morning and it was cordoned off and we’ve been moved to a different area,” Mr Freeland said.
In June 2000, the bodies of 58 Chinese immigrants were found in the back of a lorry in Dover. A Dutch lorry driver was jailed the following year for their manslaughter.