Polish and Russian officials said no-one survived after the plane apparently hit trees as it approached Smolensk airport in thick fog.
Russian media reports said the pilots ignored advice from air traffic control to divert to another airport.
Poland's army chief, central bank governor, MPs and leading historians were among more than 80 passengers.
Prime Minister Donald Tusk said the crash was the most tragic event of the country's post-World War II history.
The Polish delegation was flying in from Warsaw to mark the 70th anniversary of the Katyn massacre of thousands of Poles by Soviet forces during WWII.
Lech Kaczynski, file image
Obituary: Lech Kaczynski
In pictures: Plane crash
The BBC's Adam Easton, in Warsaw, says the crash is a catastrophe for the Polish people.
He says Prime Minister Tusk was reportedly in tears when he was told.
After an emergency meeting of ministers, Mr Tusk, who runs the day-to-day business of government, said a week of national mourning had been declared with two minutes of silence on Sunday at midday.
Mr Tusk added: "The Polish state must function and will function".
Flowers and candles laid outside presidential palace in Warsaw - 10 April 2010
Thousands have gathered outside the presidential palace in Warsaw
A government spokesman said that according to the constitution there would be an early presidential election, and the speaker of the lower house of parliament, Bronislaw Komorowski, would be acting president.
In Warsaw, people gathered outside the presidential palace to lay flowers and light candles.
"I'm all broken up... it cannot be expressed in words," Ewa Robaczewska told Reuters news agency.
The Russian emergencies ministry told Itar-Tass news agency the plane crashed at 1056 Moscow time (0656 GMT) as it was coming in to land.
Smolensk regional governor Sergei Antufiev told Russian TV that no-one had survived.
BBC Warsaw correspondent Adam Easton
Adam Easton, BBC News, Warsaw
Thousands of people have gathered outside the presidential palace to pay their respects.
There has been a spontaneous outpouring of grief, no matter what people thought of Lech Kaczynski. He was a divisive figure in Polish society, especially among younger Poles.
People are just stunned, visibly moved and in tears, whether they agreed with the president's political views or not.
The largest church bell in Poland, at Krakow Cathedral, has been rung.
It never tolls generally, only for very, very solemn occasions. The last time it did so was for the death of the Polish pope, John Paul II, five years ago.
"According to preliminary reports, it got caught up in the tops of trees, fell to the ground and broke up into pieces," he said. "There are no survivors in that crash."
Polish TV worker Slawomir Wisniewski said he had seen the crash from his hotel near the airport.
"I saw through the fog, the aeroplane flying very low with the left wing pointing to the ground," he said.
"I heard something being broken and then that thudding sound. Two flashes of fire next to each other."
Russian media carried claims that the plane's crew were at fault for the crash.
"Flight controllers... suggested that the plane be forwarded to Minsk but as far as we know the crew took an independent decision to land the plane in Smolensk," Smolensk regional government spokesman Andrei Yevseyenkov told Russian TV.
Russian officials said 97 people were killed in the crash, including eight crew.
Polish officials said that 89 people had been scheduled to fly in the delegation to the Katyn commemoration, but one person missed the flight.
Russian Prime Minister Vladimir Putin visited the crash site, after saying he would personally oversee the investigation into the crash.
"Everything must be done to establish the reasons for this tragedy in the shortest possible time," he said.
He was to meet his Polish counterpart, Mr Tusk, in Smolensk.
Russian officials said all the bodies had been recovered from the scene and were being taken to Moscow for identification.
Russia's Emergency Minister Sergei Shoigu said both of the plane's flight information recorders had been found and were being examined.
The president was flying in a Tupolev 154, a Soviet-designed plane that was more than 20 years old.
SENIOR FIGURES KILLED
President Lech Kaczynski and wife Maria
Wladyslaw Stasiak chief of the president's chancellery; Aleksander Szczyglo chief of the National Security Office; Slawomir Skrzypek National Bank of Poland chairman;
Jerzy Szmajdzinski deputy speaker of the lower house; Andrzej Kremer Foreign Ministry's undersecretary of state; Stanislaw Komorowski deputy minister of national defence; Przemyslaw Gosiewski Law and Justice party deputy chair;
Franciszek Gagor chief of the General Staff
Andrzej Przewoznik head of Poland's Council for the Protection of Struggle and Martyrdom Sites; Tomasz Merta chief historical conservator
Source: TVP1, Warsaw
Senior Polish figures killed in crash
World reaction to crash
Focus on ageing plane
Our correspondent says there had been calls for Polish leaders to upgrade their planes.
Mr Kaczynski himself had suffered scares while using the plane in late 2008, when problems with the aircraft's steering mechanism delayed his departure from Mongolia.
"Any flight brings with it a certain risk, but a very serious risk attaches to the responsibilities of a president, because it is necessary to fly constantly," he was quoted as saying at the time.
But the head of Russia's Aviakor aviation maintenance company told Russian TV the plane was airworthy, after his plant fully overhauled it in December.
As well as the president and his wife, Maria, a number of senior officials were on the passenger list.
They included the army chief of staff Gen Franciszek Gagor, central bank governor Slawomir Skrzypek and deputy Foreign Minister Andrzej Kremer.
World leaders including Mr Putin, US President Barack Obama, German Chancellor Angela Merkel and UK Prime Minister Gordon Brown offered their condolences to Poland.
Mr Kaczynski's twin brother, Jaroslaw, a former prime minister and now head of the main opposition party, was said to be "devastated", an aide told AFP news agency.
Lech Kaczynski, who had fewer powers than the prime minister but had a significant say in foreign policy, was a controversial figure in Polish politics.
He had advocated a right-wing Catholic agenda, opposed rapid free-market reforms and favoured retaining social welfare programmes.