Best v the best
"You are the best at tennis" v "You are best at tennis"
These mean the same, although both of them have a range of meanings. They could mean that you're better at tennis than other people in the room, or on the team, or at your school, or in the world. Alternatively, they could mean that you're better at tennis than at any of the other sports you play - without specifying that you're better at tennis than other people.
"You are the best" v "You are best"
If the statement was made in the context of a particular discussion (for example, about tennis), the two would have the same meaning (and the same range of meanings that we saw in the previous examples).
However, "You're the best!" as a complete sentence can also be an expression of gratitude, meaning "You're awesome!" - whereas "You're best" rarely if ever has this meaning.
"Choose the book you like the best." "Choose the book you like best."
These mean the same.
"Choose the book you like the most." "Choose the book you like most".
These mean the same.
"She walks most gracefully." v "She walks the most gracefully."
"She walks the most gracefully" usually means that she walks more gracefully than other people (although which particular group of other people is ambiguous or dependent on context, as with the tennis example). Alternatively, it could mean that she walks more gracefully than she performs other activities - this is unusual, but would be clear from the context.
"She walks most gracefully" could be a synonym for "She walks very gracefully". But "she walks most gracefully" could also be used to mean "she walks the most gracefully". So, the version without the "the" carries both meanings (or sets of meanings).