On a scheme, being normal means that each stalk of the structure sheaf is a integrally closed domain. Being regular means that each stalk of the structure sheaf is a regular local ring.

As for a local ring, being regular or being integrally closed does not imply another.

What is their connection with each other and classical/usual intuition of being smooth(being regular on stalk of each closed points)?

Moreover, is there a smooth/regular variety which is not normal?