Can someone explain the different types of unix domain sockets

Since kernel 3.3, it is possible using ss or lsof-4.89 or above — see Stéphane Chazelas's answer.

In older versions, according to the author of lsof, it was impossible to find this out: the Linux kernel does not expose this information. Source: 2003 thread on comp.unix.admin.

The number shown in /proc/$pid/fd/$fd is the socket's inode number in the virtual socket filesystem. When you create a pipe or socket pair, each end successively receives an inode number. The numbers are attributed sequentially, so there is a high probability that the numbers differ by 1, but this is not guaranteed (either because the first socket was N and N+1 was already in use due to wrapping, or because some other thread was scheduled between the two inode allocations and that thread created some inodes too).

I checked the definition of socketpair in kernel 2.6.39, and the two ends of the socket are not correlated except by the type-specific socketpair method. For unix sockets, that's unix_socketpair in net/unix/af_unix.c.

With netcat-openbsd, there is a -U option. If you don't have it, you probably have netcat-traditional installed instead; I'd suggest switching.

Example command: nc -U /var/run/socket