If computers start counting at 0, why does the init process have a pid of 1?

Processes need to have a parent (PPID). The kernel, despite not being a real process, is nevertheless handcrafting some real processes like at least init, and is giving itself the process ID 0. Depending on the OS it might or might not be displayed as a process in ps output but is always displayed as a PPID:

eg on Linux:

$ ps -ef|head
UID        PID  PPID  C STIME TTY          TIME CMD
root         1     0  0 09:09 ?        00:00:00 /sbin/init
root         2     0  0 09:09 ?        00:00:00 [kthreadd]
root         3     2  0 09:09 ?        00:00:00 [ksoftirqd/0]
...

on Solaris:

$ ps -ef|head
     UID   PID  PPID   C    STIME TTY         TIME CMD
    root     0     0   0   Oct 19 ?           0:01 sched
    root     5     0   0   Oct 19 ?          11:20 zpool-rpool1
    root     1     0   0   Oct 19 ?           0:13 /sbin/init
    root     2     0   0   Oct 19 ?           0:07 pageout
    root     3     0   1   Oct 19 ?         117:10 fsflush
    root   341     1   0   Oct 19 ?           0:15 /usr/lib/hal/hald --daemon=yes
    root     9     1   0   Oct 19 ?           0:59 /lib/svc/bin/svc.startd
...

Note also that pid 0 (and -1 and other negative values for that matter) have different meanings depending on what function use them like kill, fork and waitpid.

Finally, while the init process is traditionally given pid #1, this is no more the case when OS level virtualization is used like Solaris zones, as there can be more than one init running:

$ ps -ef|head
     UID   PID  PPID   C    STIME TTY         TIME CMD
    root  4733  3949   0 11:07:25 ?           0:26 /lib/svc/bin/svc.configd
    root  4731  3949   0 11:07:24 ?           0:06 /lib/svc/bin/svc.startd
    root  3949  3949   0 11:07:14 ?           0:00 zsched
  daemon  4856  3949   0 11:07:46 ?           0:00 /lib/crypto/kcfd
    root  4573  3949   0 11:07:23 ?           0:00 /usr/sbin/init
  netcfg  4790  3949   0 11:07:34 ?           0:00 /lib/inet/netcfgd
    root  4868  3949   0 11:07:48 ?           0:00 /usr/lib/pfexecd
    root  4897  3949   0 11:07:51 ?           0:00 /usr/lib/utmpd
  netadm  4980  3949   0 11:07:54 ?           0:01 /lib/inet/nwamd

There are two tasks with specially distinguished process IDs: swapper or sched has process ID 0 and is responsible for paging, like jlliagre put in the examples before, and is actually part of the kernel rather than a normal user-mode process.

Process ID 1 is usually the init process primarily responsible for starting and shutting down the system. Originally, process ID 1 was not specifically reserved for init by any technical measures: it simply had this ID as a natural consequence of being the first process invoked by the kernel. More recent Unix systems typically have additional kernel components visible as 'processes', in which case PID 1 is actively reserved for the init process to maintain consistency with older systems.

In general, 0 is often used to signify a 'null reference'. This meant that even though the 0 value exists, you might not use it because you want zero to signify a special value.