extract string between double quotes

I am not sure how you want to input the string. This has the effect you want to achieve, but it might need to be modified according to how the string is entered:

aa() { echo $3 ; } ; aa "abcd efgh" "ijkl mnop" "qrst uvwxyz"

Edit: So, if it is in variable (it has to be defined with quoted ") :

AA="\"abcd efgh\" \"ijkl mnop\" \"qrst uvwxyz\""
echo $AA

then:

FIRST=`echo $AA| awk -F \" '{print $2}'`
SECOND=`echo $AA| awk -F \" '{print $4}'`
THIRD=`echo $AA| awk -F \" '{print $6}'`
echo $FIRST : $SECOND : $THIRD

as jasonwryan pointed out above. You said, you wanted to use sed, but it makes it unnecessary complex :

FIRST=`echo $AA| sed 's/^\"\([^\"]*\)\".*/\1/'`
SECOND=`echo $AA| sed 's/^\"[^\"]*\" \"\([^\"]*\)\".*/\1/'`
THIRD=`echo $AA| sed 's/^\"[^\"]*\" \"[^\"]*\" \"\([^\"]*\)\".*/\1/'`

Edit2: It is actually possible to achieve completely without sed,awk,perl,.. only with bash, using its "read" builtin function like this (echos are for debugging):

#!/bin/bash

aa() {
echo '$1'="$1"
IFS=\" read aaa FIRST bbb SECOND ccc THIRD ddd <<< "$1"
echo FIRST=$FIRST : SECOND=$SECOND : THIRD=$THIRD
}

AA="\"abcd efgh\" \"ijkl mnop\" \"qrst uvwxyz\""
echo '$AA'="$AA"
aa "$AA"

You said this in a comment:

above string is in variable. I need to extract each section(first,second,third) and store it different variables

So let's just split it.

IFS=\"                  #set the shell's field separator
set -f                  #don't try to glob 
set -- $var             #split on $IFS
var1=$2 var2=$4 var3=$6 #yay
unset IFS               #restore something like a sane default

That won't handle strings which might contain backslash-escaped quotes, though. I hope that's not a problem, because I don't like doing those.

Building on (and somewhat simplifying) ludvik02's sed answer:

AA='"abcd efgh" "ijkl mnop" "qrst uvwxyz"'
AA1=$(echo "$AA" | sed -r 's/^([^"]*"){1}([^"]*).*/\2/')
AA2=$(echo "$AA" | sed -r 's/^([^"]*"){3}([^"]*).*/\2/')
AA3=$(echo "$AA" | sed -r 's/^([^"]*"){5}([^"]*).*/\2/')
(Note that                        this ↑                        is different on every line.)

The -r option to sed enables extended regular expressions.  We need that to use {n}, which means n occurrences of the preceding regex.  ([^"]*") is a compound (group) regex that matches any number of characters other than ", terminated by a ".  The example input string can match this regex up to six times (because it has six " characters); with ...s inserted to distinguish the gaps between the strings, those occurrences are

"abcd efgh" ... "ijkl mnop" ... "qrst uvwxyz"
↑↑--------↑↑----↑↑--------↑↑----↑↑----------↑
1    2       3       4       5        6

Matching any odd number (e.g., three) of occurrences of that consumes everything up through and including the nth (e.g., third) " character, i.e., the " character at the beginning of the ((n+1)/2)th (e.g., second) quoted string.  Then ([^"]*) matches (and groups) everything up to (but not including) the (n+1)th " character, i.e., the " character at the end of the second quoted string.  So this group (the second group) matches the second quoted string.  Finally, .* consumes the rest of the input string.  So then we replace the entire input line with \2, the value of the second group:

"abcd efgh" ... "ijkl mnop" ... "qrst uvwxyz"
↑↑--------↑↑----↑⇑=======⇑⇈≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡⇈
1    2       3
↑---------------↑⇑=======⇑⇈≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡⇈
        \1          \2          (unnamed)