How to find out which (not installed) Debian package a file belongs to?

apt-file

apt-file provides the feature of searching for a package providing a binary (like Debian or Ubuntu), it is not installed by default but in the repositories.

apt-file search <path-to-file>

You may want to update once before searching...

apt-file update 

For example, let's search for the not installed binary mysqldump:

$ apt-file search /usr/bin/mysqldump

mysql-client-5.1: /usr/bin/mysqldump
mysql-client-5.1: /usr/bin/mysqldumpslow
mysql-cluster-client-5.1: /usr/bin/mysqldump
mysql-cluster-client-5.1: /usr/bin/mysqldumpslow

It's also possible to list the contents of a (not-installed) package:

$ apt-file list mysql-client-5.1

mysql-client-5.1: /usr/bin/innochecksum
mysql-client-5.1: /usr/bin/innotop
mysql-client-5.1: /usr/bin/myisam_ftdump
mysql-client-5.1: /usr/bin/mysql_client_test
...

yum

yum accepts the command whatprovides (or provides) to search for installed or not installed binaries:

yum whatprovides <path-to-file>

Again, the not installed mysqldump:

$ yum whatprovides /usr/bin/mysqldump

mysql-5.1.51-2.fc14.i686 : MySQL client programs and shared libraries
Repo        : fedora
Matched from: 
Filename    : /usr/bin/mysqldump

mysql-5.1.51-1.fc14.i686 : MySQL client programs and shared libraries
Repo        : fedora
Matched from: 
Filename    : /usr/bin/mysqldump

zypper

zypper's search command can check file lists when used with the -f option.

zypper se -f /bin/mksh
Loading repository data...
Reading installed packages...

S | Name | Summary           | Type   
--+------+-------------------+--------
  | mksh | MirBSD Korn Shell | package

Webpin provides a webbased solution, there is even a script for the command-line.

pkgfile

Available as pkgtools for pacman based systems. Provides a similar search feature like the others above:

$ pkgfile -si /usr/bin/mysqldump

Name      : mysql-clients
Version   : 5.1.54-1
Url       : http://www.mysql.com/
License   : GPL
Depends   : libmysqlclient
...

The standard tool for this is apt-file. Run apt-file update to download the index file. Here's the output:

Downloading complete file ftp://ftp.is.co.za/debian/dists/squeeze/Contents-i386.gz
  % Total    % Received % Xferd  Average Speed   Time    Time     Time  Current
                                 Dload  Upload   Total   Spent    Left  Speed
100 16.6M  100 16.6M    0     0  33191      0  0:08:47  0:08:47 --:--:-- 38951

After that, run apt-file search search_term.

If you don't have access to the machine or can't use the commands given in the previous answers, you can also find this out using the Debian package web-based search engine (the second form, "Search the contents of packages").