The rc0.d,rc1.d,... directories in /etc

As others have noted, the answer is all about runlevels which are basically the modes of operation of an operating system. On Linux, these are usually:

ID  Name                               Description
0   Halt                               Shuts down the system.
1   Single-user Mode                   Mode for administrative tasks.
2   Multi-user Mode                    Does not configure network interfaces and 
                                       does not export networks services.
3   Multi-user Mode with Networking    Starts the system normally.
4   Not used/User-definable            For special purposes.
5   Start the system normally with 
    with GUI                           As runlevel 3 + display manager.
6   Reboot                             Reboots the system.

So, each of the rcN directories contains symbolic links to the scripts that should be run at that runlevel. All the actual scripts are normally in the /etc/init.d directory:

$ ls -l /etc/rc5.d/S22cron
lrwxrwxrwx 1 root root 14 Jan 14  2013 /etc/rc5.d/S22cron -> ../init.d/cron

Symbolic link naming

A symbolic link whose name begins with an S will be started at the runlevel in question while those whose name begins with K will be killed. Notice that all the links in rc6.d, the reboot runlevel, start with K. That's because they should all be stopped for a reboot and nothing should be started.

The numbers after the initial letter refer to the running order of the linked scripts. Those with smaller numbers will be run before those with higher numbers. So, in your specific example, S23ntp will be run (started in this case) before S24dovecot.

1. These are runlevels and are a System V-style initiation used by most *NIX systems (with the notable exception of systemd-based systems). When booting the kernel/user decides what runlevel should it run and execute only that runlevel. Meaning that depending the runlevel you can boot up with a different set of programs. There are runlevels for halt and reboot too, but since you are focusing on the startup part, let's ignore them for now.
2. Since only one runlevel is executed at boot, some programs should/want to start/stop at different runlevels with different or same parameters in the same or different order (not all runlevels are the same in all OS's). But Ubuntu copy runlevels 3-5 from 2, that's why they are the same.
3. No. runlevels are executed just once in startup or when you change runlevel.
4. ntp scripts should execute first then dovecot in runlevel 2-5, not the case for runlevel 1. The ordinal number in the script names (S23ntp) states the order of execution. So, it all depends of the runlevel you are using.
5. It depends of the Distro but in the particular case of Ubuntu you can add your script to runlevel 1 and 2.

More info in the Wikipedia article about Ubuntu runlevels

1) The multiple rcX.d directories specify what services to start or stop during the 'X' runlevel.

2) rc0.d is for runlevel 0 which is shutdown. rc6.d is for reboot. Rest all are for different runlevels (2 - 5). The S stands for start and K for Kill. These are essentially links to the original scripts in /etc/rc.d. The numbers after S/K are the priority by which the services will be started/Killed.

3) Yes if they are specified multiple times the start/kill script will be run multiple times. But no one wants to do that.

4) Looking at the priority numbers, ntp service will be started first followed by dovecot.

5) The 4th point's the way.