Traditional DES scheme in Unix crypt function
As for the salt, it is a two-character string chosen from the set of printable characters leading to an effective 12-bit entropy.

The fact that every eigth bit of the password is dropped is due to the DES itself: from the 64-bit key, only 56 bits are actually used, thereby dropping 8 of the bits of the key. So, you just input the 64-bit password as the encryption algorithm secret key. 

As for the all zeros block, it refers to the plaintext input to the encryption algorithm, which in the DES is a 64-bit value. 

Thus you load the password as the key, an all zeros block as the plaintext into the encryption algorithm.

Now the only thing is the encryption algorithm is a tweaked version of the DES: the salt is indeed introduced during the expansion function $E$ of the DES as described in section 4 of this paper. Instead of the standard $E$, a tweaked version $E'$ is implemented.

All of this is well described in this paper.