Long answer: Shotokai (Shoto's Association, where Shoto is the pseudonym Gichin Funakoshi used) was founded in 1935 to collect the necessary funds to build O-sensei's first Dojo (place for training, literally: "the place where enlightenment is searched"). This Dojo received the name Shotokan, the building, the house of Shoto. Those who trained there were in the end known as those that trained at the Shotokan; in time they became known as the Shotokan, even though Gichin Funakoshi never spoke of what he practiced as Shotokan, rather as Karate-do. When Gichin Funakoshi died, the heirs were Shotokai and specifically Shigeru Egami (later on Genshin Hironishi). All the Masters documents, the name Shotokai and Shotokan, his symbol (the tiger), etc., were left in his custody, Shotokai was furthermore in charge of organizing Gichin Funakoshi's funeral.
Today, the Shotokan, Hombu Dojo of Shotokai has been reconstructed together with Master Funakoshi's house and is the headquarters of Shotokai worldwide. Shotokan in Japan is thus a synonym of Shotokai.
Outside Japan, where Gichin Funakoshi's family wishes have been hard to impose, people and organizations have used the "Shotokan" name as an easy way to give themselves credibility, some have even shamelessly patented Master Gichin Funakoshi's personal insignia, obviously without either O-sensei's family consent nor Shotokai's authorization. This has obviously created a great deal of confusion among the public, to the point where some even think he created sports karate! Something he openly prohibited among his students. Gichin Funakoshi's Karate-do NEVER was sports oriented nor did it include competition, something that is uniquely Occidental.
Thus, as a conclusion, Shotokai is Shoto's organization (Gichin Funakoshi) and
the Shotokan is the Main Dojo of the Association, headquarters for Shotokai
in Japan. There is by the way one exception (another was the now defunct Shotokan
do Brazil), Shotokan Karate of America, that do have the master's authorization
to use the name Shotokan, all others are actually being disloyal to the memory
and the wishes of the founder of Karate-do.