A lottery was a game in which people could win money.
When needing to gain access to Kazran Sardick without the interference of his staff, the Eleventh Doctor arranged for them to win the lottery, even though it did not exist. The Doctor simply called them very lucky. When Kazran ordered him out, and he travelled back to Kazran's childhood, the Doctor acted as his babysitter, after his usual babysitter won the lottery, even though it did not exist, even back then. (TV: A Christmas Carol)
While working to stop a Cyberman invasion of 18th century Klimtenburg, the Eleventh Doctor was about to reassure some of the villagers that the likelihood of a Cyber guard finding them was as likely as winning the lottery. He was interrupted when the Cyber guard found them, causing him to complain that he never won the lottery unless he travelled forward in time and checked the numbers first. (PROSE: Plague of the Cybermen)
When the Eleventh Doctor met the Tenth Doctor in England, 1562, the latter compared regeneration to a lottery, implying that the former was compensating for his with his bigger sonic screwdriver. (TV: The Day of the Doctor)
In New Port City, citizens of the Grey Zone could only move to the Black Zone if they won the lottery. The lottery was played by randomising the teleportation coordinates of people's homes and selecting a winner. (PROSE: Gatecrashers)