Doctor Who has an unusual structure. It's about a time-traveling alien who brings unsuspecting humans along to have adventures and solve problems. It's a science fiction show where space battles are rarely the focal point. The Doctor loves finding unorthodox solutions to bizarre issues. Humans are more pragmatic. When we encounter aliens, we construct the Torchwood Institute to defeat them in combat and use their technology for profit.
Torchwood fared reasonably well for a Doctor Who spin-off. It enjoyed incredible viewership numbers for its first few seasons. Critics warmed to it by its third year. The show has innumerable fans, but it could never replace Doctor Who. As a standalone series, there's plenty to love about Torchwood, but it ran its natural course.
How was Torchwood formed?
The Torchwood Institute was technically created to defeat the Doctor. It was commissioned by Queen Victoria in 1879 after her first encounter with the Time Lord. The Doctor and Rose landed in Scotland, only a few feet from Victoria's carriage. She was traveling to the Torchwood Estate, a castle owned by a Scottish noble called Sir Robert MacLeish. The Doctor convinces the Queen's entourage that he's her new protector so he and Rose can join her. They travel to the Torchwood Estate and discover a small group of monks has taken over the mansion. They've captured a werewolf and intend to pass lycanthropy onto the Queen. The Doctor uses the Koh-i-Noor diamond and a novelty telescope to kill the wolf. Victoria rewards the Doctor and Rose with a knighthood before having them both exiled. Victoria decides that the Doctor, who just saved her life, threatens the British Empire. She founded an organization to stop beings like him from bothering the monarchy.
Victoria established the Torchwood Institute in memory of the late Robert MacLeish. She named the Doctor an enemy of the Crown. If the Doctor ever violated her orders and returned to Great Britain, Torchwood would be waiting. Their charter gave them broad power to act under the monarchy's name. They were expected to fight alien and supernatural threats. In 1882, Victoria gave them the additional mandate to acquire any alien technology found on Earth. The organization's goal remained defensive, but the influx of power added elements. The Torchwood Institute intended to use advanced tech to re-establish the British Empire. This never panned out, but it's worth keeping in mind. Torchwood's motto became "If it's alien, it's ours."
How did Jack Harkness get involved with Torchwood?
Everyone's favorite member of the organization remains this immortal omnisexual. Captain Jack is best known as an agent of Torchwood Three, an off-shoot of the original organization. In 1885, Torchwood discovered a rift in space-time above Cardiff. Torchwood Three was created to monitor and address the mysterious anomaly. Agents Emily Holroyd and Alice Guppy worked for Torchwood Three. In 1899, they discovered Captain Jack. They kept him under surveillance after they overheard him talking about the Doctor. Alice and Emily had Jack captured and interrogated. After learning he couldn't die, they coerced him into service for over a century. Jack became the leader of Torchwood Three in 2000, though there were almost always multiple iterations of the man on Earth.
What happened to Torchwood?
Torchwood was irreparably damaged in the Battle of Canary Wharf. The organization deserved its fate. Torchwood One attempted to use a mysterious energy in the air above Canary Wharf as fuel. They planned to overcome their reliance on foreign oil and reinstate the British Empire with a firmer grip on power. Torchwood unintentionally weakened the boundaries between dimensions, allowing Cybermen and Daleks to invade Earth simultaneously. Tons of Torchwood agents were killed or converted into Cybermen. Naturally, the Doctor, Torchwood's first target, solved the problem for them by sealing all the aliens in an alternate dimension. Jack Harkness worked to stop an alien called the 456 from killing millions of children, leaving him one of two remaining Torchwood members. After Jack returned from an off-world vacation, he found Gwen Cooper and Rhys Williams reforming Torchwood. The organization was publically known after Canary Wharf. They took the name to form a new group dedicated to the lessons the Doctor left behind.
Torchwood spent most of its time as a sinister organization that demonstrated what humans would do when faced with interstellar oddities. Leave it to the British to learn of life on other planets and declare it an enemy of the Crown. Though their charter is questionable, Torchwood always had decent people trying to do the right thing in their ranks. They made mistakes because their initial creation was in service of something pointless and bad. By the 2010s, Torchwood died and revived as something worthwhile. Even in the most questionable government organizations, good people ignore the top brass and do their best to help others.