Generally, to go through a list, keeping only items which meet a condition. The word originated with the ed editor, in which the command 'g/re/p', where 're' is a regular expression, would globally (that's the 'g') search for lines matching the regular expression in the file you were editing and print (that's the 'p') the results.
Presumably some time after ed was written, the program grep was born. grep, some version of which can be found on virtually every UNIX installation, searches through files and prints lines matching a regular expression. The grep program embodies the UNIX philosophy: a small, simple, reusable tool that handles plain text. grep is flexible: among the many options you can give it, you can tell it to print non-matching lines (-v), print some number of lines before (-B), after (-A) or around (-C (context)) matches, to count the number of matches (-c), ignore case (-i), print line numbers (-n), and to search recursively through directories (-r). grep is perfect for use in pipe-linked shell commands.
Examples:
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grep -n foo somefile
Searches for the pattern "foo" in somefile, and prints lines found, with line numbers.
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grep '^.b.$' /usr/share/dict/words
Searches for three-letter words with the middle letter "b" in the dictionary: this kind of thing is great for cheating at crosswords.
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last | grep daf | head
Lists my last ten logins.
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find /var/log/apache | grep 'access.*gz$' | xargs zcat | grep ja.net | awk '{print $1}' | sort | uniq
Finds compressed Apache log files, get the contents, search for accesses from ja.net, extract the hostname, sort the results and remove duplicates. I used this, out of curiosity, to get a list of the JANET proxy servers that had accesed a web server, since they are named after pizza toppings. grep is used here twice: first to select filenames and secondly to select lines from those files.
See your local grep manual page for details, or see the GNU documentation for their version at http://www.gnu.org/software/grep/grep.html.
Since grep's invention, however, the verb "grep" has acquired a more general meaning: to search through a list of things, keeping only items which meet a certain condition. Some programming languages, notably Perl, have a grep function, a higher-order function that selects values from a list. Each value is only kept if substituting that value into an expression returns a true value. Other languages have an equivalent function, such as Scheme's select, Ruby's Array#select, and Haskell's filter.
grep can be defined nicely in Haskell as
grep predicate list = [ item | item <- list; predicate item ]
Examples:
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Perl
grep { $_ > 3 } qw(1 2 3 4 5 6);
# result: qw(4 5 6)
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Ruby
[1, 2, 3, 4, 5, 6].select { |x| x > 3 }
# result: [4, 5, 6]
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Haskell
filter (>3) [1, 2, 3, 4, 5, 6]
-- result: [4, 5, 6]
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Scheme
(select (lambda (x) (> x 3)) '(1 2 3 4 5 6))
; result: '(4 5 6)