Eon Productions has been around for eons, just like James Bond. They formed in 1961 for the sole purpose of bringing Bond to the world, and they've been at it ever since.
However, it took them a while to say "yes" to Dr. No. The latest Bond novel at the time was Thunderball, and Cubby Broccoli and Harry Saltzman, founders of Eon, wanted to bring that novel to the big screen as Bond's big ballsy debut. They even had Richard Maibaum write a draft before casting Connery as Bond. (Source)
Legal trouble put the kibosh on that production. Yes, Bond was defeated by a lawyer before he even got started.
Broccoli and Saltzman (which is coincidentally what we had for dinner last night) wanted to adapt the first Bond novel, Casino Royale, instead, but those rights had already been purchased by a man named Charles K. Feldman. After Bond became a hit, Feldman would produce a parody version of Casino Royale in 1967 starring David Niven and Bond girl Ursula Andress. (Source)
So they decided on Dr. No, but don't tell him that they settled. Broccoli and Saltzman treated No as if it was their first choice, their favorite child, the yummiest French fry in the box. By doting on it, they laid the foundation for one of the longest-running film series of all time.
And they eventually got to make the other movies too. Thunderball became the fourth Bond in 1965, and the official Eon-produced Casino Royale introduced Daniel Craig as Bond in 2006, eons after Bond's debut in Dr. No.