Quaternions are used to represent rotations.

They are compact, don't suffer from gimbal lock and can easily be interpolated.
Unity internally uses Quaternions to represent all rotations.

They are based on complex numbers and are not easy to understand intuitively.
You almost never access or modify individual Quaternion components (x,y,z,w);
most often you would just take existing rotations (e.g. from the Transform) and use them
to construct new rotations (e.g. to smoothly interpolate between two rotations).
The Quaternion functions that you use 99% of the time are:
Quaternion.LookRotation, Quaternion.Angle, Quaternion.Euler, Quaternion.Slerp, Quaternion.FromToRotation, and Quaternion.identity. (The other functions are only for exotic uses.)

You can use the Quaternion.operator * to rotate one rotation by another, or to rotate a vector by a rotation.

identity | The identity rotation (Read Only). |

eulerAngles | Returns the euler angle representation of the rotation. |

this[int] | Access the x, y, z, w components using [0], [1], [2], [3] respectively. |

w | W component of the Quaternion. Don't modify this directly unless you know quaternions inside out. |

x | X component of the Quaternion. Don't modify this directly unless you know quaternions inside out. |

y | Y component of the Quaternion. Don't modify this directly unless you know quaternions inside out. |

z | Z component of the Quaternion. Don't modify this directly unless you know quaternions inside out. |

Quaternion | Constructs new Quaternion with given x,y,z,w components. |

Set | Set x, y, z and w components of an existing Quaternion. |

SetFromToRotation | Creates a rotation which rotates from fromDirection to toDirection. |

SetLookRotation | Creates a rotation with the specified forward and upwards directions. |

ToAngleAxis | Converts a rotation to angle-axis representation. |

ToString | Returns a nicely formatted string of the Quaternion. |

Angle | Returns the angle in degrees between two rotations a and b. |

AngleAxis | Creates a rotation which rotates angle degrees around axis. |

Dot | The dot product between two rotations. |

Euler | Returns a rotation that rotates z degrees around the z axis, x degrees around the x axis, and y degrees around the y axis (in that order). |

FromToRotation | Creates a rotation which rotates from fromDirection to toDirection. |

Inverse | Returns the Inverse of rotation. |

Lerp | Interpolates between from and to by t and normalizes the result afterwards. |

LookRotation | Creates a rotation with the specified forward and upwards directions. |

RotateTowards | Rotates a rotation from towards to. |

Slerp | Spherically interpolates between from and to by t. |

operator != | Are two quaternions different from each other? |

operator * | Combines rotations lhs and rhs. |

operator == | Are two quaternions equal to each other? |