The question asks how to check if a variable is an empty string and the best answers are already given for that.
But I landed here after a period passed programming in php and what I was actually searching was a check like the empty function in php working in a bash shell.
After reading the answers I realized I was not thinking properly in bash, but anyhow in that moment a function like empty in php would have been soooo handy in my bash code.
As I think this can happen to others, I decided to convert the php empty function in bash

According to the php manual:
a variable is considered empty if it doesn't exist or if its value is one of the following:

• "" (an empty string)
• 0 (0 as an integer)
• 0.0 (0 as a float)
• "0" (0 as a string)
• an empty array
• a variable declared, but without a value

Of course the *null* and *false* cases cannot be converted in bash, so they are omitted.

function empty
{
    local var="$1"
    
    # Return true if:
    # 1.    var is a null string ("" as empty string)
    # 2.    a non set variable is passed
    # 3.    a declared variable or array but without a value is passed
    # 4.    an empty array is passed
    if test -z "$var"
    then
        [[ $( echo "1" ) ]]
        return
    
    # Return true if var is zero (0 as an integer or "0" as a string)
    elif [ "$var" == 0 2> /dev/null ]
    then
        [[ $( echo "1" ) ]]
        return
        
    # Return true if var is 0.0 (0 as a float)
    elif [ "$var" == 0.0 2> /dev/null ]
    then
        [[ $( echo "1" ) ]]
        return
    fi
    
    [[ $( echo "" ) ]]
}

Example of usage:

if empty "${var}"
    then
        echo "empty"
    else
        echo "not empty"
fi

Demo:
the following snippet:

#!/bin/bash

vars=(
    ""
    0
    0.0
    "0"
    1
    "string"
    " "
)

for (( i=0; i<${#vars[@]}; i++ ))
do
    var="${vars[$i]}"
    
    if empty "${var}"
        then
            what="empty"
        else
            what="not empty"
    fi
    echo "VAR \"$var\" is $what"
done

exit

outputs:

VAR "" is empty
VAR "0" is empty
VAR "0.0" is empty
VAR "0" is empty
VAR "1" is not empty
VAR "string" is not empty
VAR " " is not empty