What is the equivalent of Active Directory on Linux

You either build your own Active Directory-equivalent from Kerberos and OpenLDAP (Active Directory basically is Kerberos and LDAP, anyway) and use a tool like Puppet (or OpenLDAP itself) for something resembling policies, or you use FreeIPA as an integrated solution.

There's also a wide range of commercially supported LDAP servers for Linux, like Red Hat Directory Server. RHDS (like 389 Server, which is the free version of RHDS) has a nice Java GUI for management of the directory. It does neither Kerberos nor policies though.

Personally, I really like the FreeIPA project and I think it has a lot of potential. A commercially supported version of FreeIPA is included in standard RHEL6 subscriptions, I believe.

That said, what your are asking about is more like a fileserver solution than an authentication solution (which is what AD is). If you want your files on all machines you log into, you have to set up an NFS server and export an NFS share from your fileserver to your network. NFSv3 has IP-range based ACL's, NFSv4 would be able to do proper authentication with Kerberos and combines nicely with the authentication options I described above.

If you have Windows boxes on your network, you will want to setup a Samba server, which can share out your files to Linux and Windows boxes alike. Samba3 can also function as an NT4 style domain controller, whereas Samba4 is able to mimic a Windows 2003 style domain controller.

If you just want centralized authentication, look at NIS or NIS+ (formerly known as yellow pages which is why all the commands begin with 'yp').

Configure your main server as the master NIS server, then configure all the other boxes to use NIS to authenticate users.

The wikipedia page for NIS and the Linux NIS Howto.

For a basic home network NIS will be fine. If you need more control over which users can see which servers you'll need to use NIS+.