A Guessing Game

Let's play a little game to give you an idea of how different algorithms for the same problem can have wildly different efficiencies. The computer is going to randomly select an integer from 1 to 16. You have to guess the number by making guesses until you find the number that the computer chose. Let's begin.

Maybe you guessed $1$, then $2$, then $3$, then $4$, and so on, until you guessed the right number. We call this approach linear search, because you guess all the numbers as if they were lined up in a row. It would work. But what is the highest number of guesses you could need? If the computer selects $16$ ...