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They are the building blocks for all numbers just as letters are the building blocks for words.

Number theory is the study of ( among other things) primes and how they behave.

the Fundamental Theorem of Arithmetic (or Unique-Prime-Factorization Theorem) states that any integer greater than 1 can be written as a unique product (up to ordering of the terms) of prime numbers. This is one of the most important theorems in all of mathematics.

Another important thing to mention about primes is the Euclid proved there is an infinite number of them.

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Q: How has prime numbers change math?
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