On Earth, the 20th century was one of the most turbulent, violent, yet progressive centuries in the planet's history. Between major conflicts including the First and Second World Wars, humankind began to explore beyond their homeworld for the first time, and also began to make huge leaps in technological advancement. It was also during the 20th century that many world governments officially learned of the existence of extraterrestrial lifeforms (although few in the general public would know of this until the beginning of the 21st century). The Torchwood Institute, founded in the 19th century, began to gain power and influence in this century, while several attemped alien invasions resulted in the formation of the United Nations Intelligence Taskforce.
The Doctor had a great affinity for this century. He once claimed to tolerate, but not enjoy it. (DW: An Unearthly Child) He lived during this time period for several years as an exile from Gallifrey (during his third incarnation) and again for a number of months in 1913 and 1969 in his tenth incarnation (to name just two examples). Many of the Doctor's known Earth companions came from this century.