What does the number in parentheses shown after Unix command names in manpages mean?

For example: man(1), find(3), updatedb(2)?

What do the numbers in parentheses (Brit. "brackets") mean?

It's the section that the man page for the command is assigned to.

These are split as

1. General commands
2. System calls
3. C library functions
4. Special files (usually devices, those found in /dev) and drivers
5. File formats and conventions
6. Games and screensavers
7. Miscellanea
8. System administration commands and daemons

Original descriptions of each section can be seen in the Unix Programmer's Manual (page ii).

In order to access a man page given as "foo(5)", run:

man 5 foo

The section the command is documented in the manual. The list of sections is documented on man's manual. For example:

man 1 man
man 3 find

This is useful for when similar or exactly equal commands exist on different sections

The reason why the section numbers are significant is that many years ago when disk space was more of an issue than it is now the sections could be installed individually.

Many systems only had 1 and 8 installed for instance. These days people tend to look the commands up on google instead.

As @Ian G says, they are the man page sections. Let's take this one step further though:

1. See the man page for the man command with man man, and it shows the 9 sections as follows:

DESCRIPTION
       man  is  the system's manual pager. Each page argument given
       to man is normally the name of a program, utility  or  func‐
       tion.   The  manual page associated with each of these argu‐
       ments is then found and displayed. A section,  if  provided,
       will  direct man to look only in that section of the manual.
       The default action is to search in all of the available sec‐
       tions following a pre-defined order ("1 n l 8 3 2 3posix 3pm
       3perl 5 4 9 6 7" by default, unless overridden by  the  SEC‐
       TION directive in /etc/manpath.config), and to show only the
       first page found, even if page exists in several sections.

       The table below shows the section numbers of the manual fol‐
       lowed by the types of pages they contain.

       1   Executable programs or shell commands
       2   System calls (functions provided by the kernel)
       3   Library calls (functions within program libraries)
       4   Special files (usually found in /dev)
       5   File formats and conventions eg /etc/passwd
       6   Games
       7   Miscellaneous  (including  macro  packages  and  conven‐
           tions), e.g. man(7), groff(7)
       8   System administration commands (usually only for root)
       9   Kernel routines [Non standard]

       A manual page consists of several sections.

2. man <section_num> <cmd>

Let's imagine you are Googling around for Linux commands. You find the OPEN(2) pg online: open(2) — Linux manual page.

To see this in the man pages on your pc, simply type in man 2 open.

For FOPEN(3) use man 3 fopen, etc.

3. man <section_num> intro

To read the intro pages to a section, type in man <section_num> intro, such as man 1 intro, man 2 intro, man 7 intro, etc.

To view all man page intros in succession, one-after-the-other, do man -a intro. The intro page for Section 1 will open. Press q to quit, then press Enter to view the intro for Section 8. Press q to quit, then press Enter to view the intro for Section 3. Continue this process until done. Each time after hitting q, it'll take you back to the main terminal screen but you'll still be in an interactive prompt, and you'll see this line:

--Man-- next: intro(8) [ view (return) | skip (Ctrl-D) | quit (Ctrl-C) ]

Note that the Section order that man -a intro will take you through is:

1. Section 1
2. Section 8
3. Section 3
4. Section 2
5. Section 5
6. Section 4
7. Section 6
8. Section 7

This search order is intentional, as the man man page explains:

The default action is to search in all of the available sections follow‐
ing a pre-defined order ("1 n l 8 3 2 3posix 3pm 3perl 5 4 9 6 7" by default, unless overrid‐
den  by the SECTION directive in /etc/manpath.config)

Why did they choose this order? I don't know (please answer in the comments if you know), but just realize this order is correct and intentional.

Related:

1. Google search for "linux what does the number mean in parenthesis after a function?"
2. SuperUser: What do the parentheses and number after a Unix command or C function mean?
3. Unix & Linux: What do the numbers in a man page mean?