When encrypting, you use their public key to write a message and they use their private key to read it.

When signing, you use your private key to write message's signature, and they use your public key to check if it's really yours.

I want to use my private key to generate messages so only I can possibly be the sender.

I want my public key to be used to read the messages and I do not care who reads them

This is signing, it is done with your private key.

I want to be able to encrypt certain information and use it as a product key for my software.

I only care that I am the only one who can generate these.

If you only need to know it to yourself, you don't need to mess with keys to do this. You may just generate random data and keep it in a database.

But if you want people to know that the keys are really yours, you need to generate random data, keep in it a database AND sign it with your key.

I would like to include my public key in my software to decrypt/read the signature of the key.

You'll probably need to purchase a certificate for your public key from a commercial provider like Verisign or Thawte, so that people may check that no one had forged your software and replaced your public key with theirs.

bcrypt is a hashing algorithm which is scalable with hardware (via a configurable number of rounds). Its slowness and multiple rounds ensures that an attacker must deploy massive funds and hardware to be able to crack your passwords. Add to that per-password salts (bcrypt REQUIRES salts) and you can be sure that an attack is virtually unfeasible without either ludicrous amount of funds or hardware.

bcrypt uses the Eksblowfish algorithm to hash passwords. While the encryption phase of Eksblowfish and Blowfish are exactly the same, the key schedule phase of Eksblowfish ensures that any subsequent state depends on both salt and key (user password), and no state can be precomputed without the knowledge of both. Because of this key difference, bcrypt is a one-way hashing algorithm. You cannot retrieve the plain text password without already knowing the salt, rounds and key (password). [Source]

How to use bcrypt:

Using PHP >= 5.5-DEV

Password hashing functions have now been built directly into PHP >= 5.5. You may now use password_hash() to create a bcrypt hash of any password:

<?php
// Usage 1:
echo password_hash('rasmuslerdorf', PASSWORD_DEFAULT)."\n";
// $2y$10$xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
// For example:
// $2y$10$.vGA1O9wmRjrwAVXD98HNOgsNpDczlqm3Jq7KnEd1rVAGv3Fykk1a

// Usage 2:
$options = [
  'cost' => 11
];
echo password_hash('rasmuslerdorf', PASSWORD_BCRYPT, $options)."\n";
// $2y$11$6DP.V0nO7YI3iSki4qog6OQI5eiO6Jnjsqg7vdnb.JgGIsxniOn4C

To verify a user provided password against an existing hash, you may use the password_verify() as such:

<?php
// See the password_hash() example to see where this came from.
$hash = '$2y$07$BCryptRequires22Chrcte/VlQH0piJtjXl.0t1XkA8pw9dMXTpOq';

if (password_verify('rasmuslerdorf', $hash)) {
    echo 'Password is valid!';
} else {
    echo 'Invalid password.';
}

Using PHP >= 5.3.7, < 5.5-DEV (also RedHat PHP >= 5.3.3)

There is a compatibility library on GitHub created based on the source code of the above functions originally written in C, which provides the same functionality. Once the compatibility library is installed, usage is the same as above (minus the shorthand array notation if you are still on the 5.3.x branch).

Using PHP < 5.3.7 (DEPRECATED)

You can use crypt() function to generate bcrypt hashes of input strings. This class can automatically generate salts and verify existing hashes against an input. If you are using a version of PHP higher or equal to 5.3.7, it is highly recommended you use the built-in function or the compat library. This alternative is provided only for historical purposes.

class Bcrypt{
  private $rounds;

  public function __construct($rounds = 12) {
    if (CRYPT_BLOWFISH != 1) {
      throw new Exception("bcrypt not supported in this installation. See http://php.net/crypt");
    }

    $this->rounds = $rounds;
  }

  public function hash($input){
    $hash = crypt($input, $this->getSalt());

    if (strlen($hash) > 13)
      return $hash;

    return false;
  }

  public function verify($input, $existingHash){
    $hash = crypt($input, $existingHash);

    return $hash === $existingHash;
  }

  private function getSalt(){
    $salt = sprintf('$2a$%02d$', $this->rounds);

    $bytes = $this->getRandomBytes(16);

    $salt .= $this->encodeBytes($bytes);

    return $salt;
  }

  private $randomState;
  private function getRandomBytes($count){
    $bytes = '';

    if (function_exists('openssl_random_pseudo_bytes') &&
        (strtoupper(substr(PHP_OS, 0, 3)) !== 'WIN')) { // OpenSSL is slow on Windows
      $bytes = openssl_random_pseudo_bytes($count);
    }

    if ($bytes === '' && is_readable('/dev/urandom') &&
       ($hRand = @fopen('/dev/urandom', 'rb')) !== FALSE) {
      $bytes = fread($hRand, $count);
      fclose($hRand);
    }

    if (strlen($bytes) < $count) {
      $bytes = '';

      if ($this->randomState === null) {
        $this->randomState = microtime();
        if (function_exists('getmypid')) {
          $this->randomState .= getmypid();
        }
      }

      for ($i = 0; $i < $count; $i += 16) {
        $this->randomState = md5(microtime() . $this->randomState);

        if (PHP_VERSION >= '5') {
          $bytes .= md5($this->randomState, true);
        } else {
          $bytes .= pack('H*', md5($this->randomState));
        }
      }

      $bytes = substr($bytes, 0, $count);
    }

    return $bytes;
  }

  private function encodeBytes($input){
    // The following is code from the PHP Password Hashing Framework
    $itoa64 = './ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789';

    $output = '';
    $i = 0;
    do {
      $c1 = ord($input[$i++]);
      $output .= $itoa64[$c1 >> 2];
      $c1 = ($c1 & 0x03) << 4;
      if ($i >= 16) {
        $output .= $itoa64[$c1];
        break;
      }

      $c2 = ord($input[$i++]);
      $c1 |= $c2 >> 4;
      $output .= $itoa64[$c1];
      $c1 = ($c2 & 0x0f) << 2;

      $c2 = ord($input[$i++]);
      $c1 |= $c2 >> 6;
      $output .= $itoa64[$c1];
      $output .= $itoa64[$c2 & 0x3f];
    } while (true);

    return $output;
  }
}

You can use this code like this:

$bcrypt = new Bcrypt(15);

$hash = $bcrypt->hash('password');
$isGood = $bcrypt->verify('password', $hash);

Alternatively, you may also use the Portable PHP Hashing Framework.