One of the major applications of prime numbers is in cryptography. For example, computer systems use prime numbers for security. By taking two very large prime numbers and multiplying them together, we get a composite number. In order to crack the code, the hacker would need to know the original two prime numbers which is extremely difficult to find since there is no pattern to prime numbers. Some worry that if a pattern to prime numbers was ever found, that we would lose this security and any system could be hacked into.
The link below shows a short YouTube clip that explains this concept more in-depth.
Prime Numbers & Public Key Cryptography