The idea was that the big guns would allow it to take out destroyers, cruisers, and other smaller ships before the battlecruiser ever got into the range of their smaller guns or torpedoes, while its speed would enable it to escape enemy battleships. The idea was mainly conceived by Admiral Jackie Fisher.
The first battlecruisers were HMS Inflexible, HMS Invincible and HMS Indomitable, all completed in 1908. They had armor 6 or 7 inches thick along the side of the hull and over the gunhouses, whereas a comparable battleship of the period had armor 11 or 12 inches thick. Originally known as battle cruisers, these ships had a top speed of 26 knots compared to 20 - 21 knots for a contemporary battleship. They were armed with 12 inch (305 mm) guns, just like battleships. Soon after the British, the Germans started building their own battlecruisers, the first was SMS Von der Tann of 1911. They had only 11 inch (280 mm) guns, but were better armoured than the British ones.
In practice, the original battlecruiser concept largely failed.
It was successful at the Battle of the Falkland Islands during World War I when the British battlecruisers HMS Inflexible and HMS Invincible annihilated a German cruiser squadron commanded by Admiral Maximilian Graf Von Spee. But as soon as battlecruisers encountered enemy battlecruisers (or more modern battleships which could keep up with them), their speed advantage would be lost and the thin armor then meant that they were highly vulnerable to their opponent's gunfire. They were also at grave risk if they tried to take on battleships.
At the Battle of Jutland two years later, the British battlecruisers engaged the German battlecruisers and battleships before the arrival of the battleships of the British Grand Fleet, with disastrous results. The battlecruisers HMS Invincible , HMS Queen Mary and HMS Indefatigable exploded with the loss of all but a handful of their crews, and HMS Lion only survived by intentionally flooding one of her magazines. The German battlecruisers were better armoured, but SMS Lutzow sunk from the damage, and SMS Seydlitz was heavily damaged. No British or German battleship was sunk during the battle, but one old pre-dreadnought. Thereafter, the Royal Navy de-emphasized battlecruisers. HMS Hood was launched in 1920, and was the last British battlecruiser to be built. Between the wars, it was the biggest warship in the world. Her armour was stronger, than of earlier battlecruisers, but it also proved a fatal weakness, as she exploded and sank in a duel with Bismarck during World War II.
Other navies persisted with the battlecruiser concept somewhat longer. The US Navy aircraft carriers USS Lexington and USS Saratoga were built on battlecruiser hulls repurposed after the Washington Naval Treaty of 1922. The US later built the battlecruiser-like "large cruisers" USS Alaska (CB-1) and Guam (CB-2), which served effectively in WWII, but a planned additional four in the class were cancelled after the war. The German Panzerschiffe (pocket battleships) (Deutschland, Admiral Scheer, and Admiral Graf Spee) were another attempt at the concept, and displayed the same weakness in the lack of armour at the Battle of the River Plate, while the German Scharnhorst and Gneisenau were labelled battlecruisers, but they traded lighter armament (11-inch main guns) rather than thinner armor for speed.
Improved engine technology also worked against the battlecruiser formula. The ultimate limit on ship speed was drag from the water displaced (which increases as a cube of speed) rather than weight, so heavier armor slowed World War II battleships by only a couple of knots over their more lightly armored brethren. As it turned out, however, aircraft carriers made both battleships and battlecruisers largely obsolete.