Wikipedia defines a spin-off to be "derived from one or more already existing works, that focuses, in particular, in more detail on one aspect of that original work (e.g. a particular topic, character, or an event)." In layman's terms, this means that a spin-off is a show about something from a popular show, usually a character. The show exist in the same universe, meaning that events (such as an invasion of aliens) or facts (such as the existance of superpowers) are the same in both shows.
For example, a Harry Potter spin-off might feature James Potter, Harry's father, as the main character. The story would explain things referenced in the books, and the same rules about magic would apply, but it's not about Harry himself. The two stories would be connected and take place in the same universe, but it's not necessarily a sequel or a prequel. Because the two stories exist in the same universe, however, their relative timeline of events must be consistent. For books, that's not too hard, but for TV shows, that's a little more complicated. Especially for Doctor Who, because it involves time travel, you can end up with diagrams like these:
Doctor Who has three main spin-offs: Class, Torchwood, and The Sarah Jane Adventures. It also had a few minor spin-offs, including K-9 and Company, K-9, and P.R.O.B.E. Be sure to check out the age rating for each show—check out the TARDIS Data Core Parent Page for more information.
What It's About: "The sixth formers of Coal Hill Academy all have their own secrets and desires. They have to deal with the stresses of everyday life, including friends, parents, school work, sex, and sorrow, but also the horrors that come from time travel. The Doctor and his time-travelling have made the walls of space and time stretch thin, and monsters beyond imagination are planning to break through and wreak havoc upon the Earth." Class revolves around an alien prince, Charlie, a refugee hiding on Earth after the destruction of his own people. His protector and former enemy, Miss Quill, pretends to be a teacher at Coal Hill, where Charlie attends. When the hostile aliens begin to attack Coal Hill, Miss Quill and Charlie, along with his friends April, Tanya, and Ram, are tasked with protecting the Earth.
Why You Should Watch It: Class is a good show, even seen by itself. It's connections with Doctor Who are, like all of its spin-offs, tenuous and subtle, except for the occasional visit from the Doctor. It's not necessarily going to add any information to the main show, but it's a good watch and an enjoyable show.
Age Rating: 12+ due to violence, gore, nudity, and sexual references.
What It's About: "This series is set inside a branch of a covert agency called the Torchwood Institute, where dashing commander in chief Capt. Jack Harkness leads Torchwood Three, an ace team of experts, as it investigates unexplained extraterrestrial incidents on Earth. Torchwood Three operates independently of the British government, law enforcement groups, and even the United Nations." After being standed by the Doctor, Captain Jack Harkness decides that the Earth needs protecting from aliens. He recruits an elite team to track down extra-terrestrial incursions and defend humanity.
Why You Should Watch It: Torchwood provides important backstory for Doctor Who, even though the Doctor never appears. It explains what happens to Captain Jack in between The Parting of the Ways and Utopia. Martha Jones, former companion of the Doctor, appears for several episodes that set up some of her scenes in Doctor Who. Most importantly, the whole Torchwood team appears in Doctor Who episodes The Stolen Earth/Journey's End. Plus, it's just a good show overall.
Age Rating: 14+ due to violence, gore, and explicit sex and nudity.
What It's About:"Investigative journalist Sarah Jane Smith, the Doctor's former traveling companion, uses her wits and some highly advanced technology to keep Earth safe from extraterrestrial intruders, assisted by her adopted (and genetically engineered) son, Luke, and her teenage neighbor, Maria Jackson." The Sarah Jane Adventures is a children's show about Sarah Jane Smith defending the Earth and fighting aliens alongside her son Luke and his friends Maria, Clyde, and Rani.
Why You Should Watch It: The Sarah Jane Adventures is just a watered-down Doctor Who. It's a show about fighting aliens, love and loss, and finding youself, just less scary and with fewer deaths. Even if you're not a child, you'll still enjoy watching it. The Doctor appears several times, in The Wedding of Sarah Jane Smith, and The Death of the Doctor, and Sarah Jane's crew appears on Doctor Who in The Stolen Earth/Journey's End.
Age Rating: 8+ for serious storylines, children's emotions, and fart jokes.
What It's About: "K9 and Company was a proposed Doctor Who spin-off in the early 1980s. Ultimately, only a single pilot episode was produced. The series was to have starred Elisabeth Sladen reprising her role as former companion Sarah Jane Smith, with John Leeson returning as the voice of K9." Sarah Jane goes to see her aunt, but finds her missing. Along with K-9 and her friend Brendan, Sarah Jane gets caught up in a mysterious cult intent on a human sacrifice.
Why You Should Watch It: Despite never being developed into a full series, K-9 and Company provides a lot of details about Sarah Jane's life without the Doctor. People and events first referenced in K-9 and Company show up later in The Five Doctors and School Reunion, and also in The Sarah Jane Adventures. It's a fun watch, and more Sarah Jane is always a plus.
Age Rating: 10+ for general scariness.
What It's About: "K-9 is set 2050 London and revolves around Professor Gryffen, who is experimenting with a Space-Time Manipulator, and his friends, 14-year-olds Starkey and Jorjie and 15-year-old Darius, who runs errands for the professor. After K-9 gets into some trouble and is saved by the team, K-9 and the humans then form the front line of defense against alien menaces from outer space and other times."
Why You Should Watch It: Because K-9 is not a BBC production, it legally connot make any references to Doctor Who. The only real relation between the shows is the character of K-9. It's also a children's show. You may want to watch it just for completions' sake.
Age Rating: 6+ for being a children's show.
What It's About: "P.R.O.B.E stars Caroline John as her Doctor Who character of Liz Shaw, who now works for the Preternatural Research Bureau." Liz Shaw works to investigate ghosts, mysterious murders, and sinister rituals.
Why You Should Watch It: P.R.O.B.E was the first ongoing Doctor Who spin-off. It provides more background to Liz Shaw. However, it was not allowed to make overt references to Doctor Who, and some information in the show contradicts what is stated in Doctor Who. You may want to watch it just for completions' sake.
Age Rating: 14+ for horror and gore.
We recommend watching Torchwood, The Sarah Jane Adventures, and Class because they have direct connections to Doctor Who. They're arguably the three best shows. The other spin-offs are fun to watch and nice for completions' sake, but not necessary for watching Doctor Who.