The UK is now at its highest threat level - meaning further attacks are expected "imminently".
Mr Brown - who was born in Glasgow - said: "The first duty of the Government is the security and safety of all the British people. So it's right to raise the levels of security at airports and crowded places. "I want all British people to be vigilant and support the police. I know the British people will stand together united, resolute and strong."
Liverpool's John Lennon airport and Blackpool airport have been closed and security tightened at other UK airports. The US has also installed extra security at its airports.
The Jeep Cherokee was rammed into the terminal at 3.11pm this afternoon. (Sky News)
**Update**
Daily Merlin Insight
Giver of Spheres - Depiction of 'the fool/jester' receiving a sphere from the giver of spheres.
Giver of Weapons - A kneeling 'knight' receiving the flaming sword from the giver of weapons.