display checkbox list on selecting value of dropdownlist

The problem is in getDays you are using a variable select but it is not defined. Solution is to pass the select instance when the onchange is called like onchange="getDays(this).

Also I think you may have to hide the checkbox list if any other option is selected.

<b>Please enter days required</b><br/>
<select name="days" id="days" style="width:200px;" onchange="getDays(this)">
    <option value="0">Select</option>
    <option value="1">Mon-Fri</option>
    <option value="2">Mon-Fri</option>
    <option value="3">Mon-Fri</option>
    <option value="4">Bespoke Days</option>
</select><br/><br/>
<div id="days_target" style="display:none;">
    <input type="checkbox" name="day" value="mon"/>Mon &nbsp;&nbsp; <input type="checkbox" name="day" value="tue"/>Tue<br/>
    <input type="checkbox" name="day" value="wed"/>Wed &nbsp;&nbsp; <input type="checkbox" name="day" value="thr"/>Thr<br/>
    <input type="checkbox" name="day" value="fri"/>Fri &nbsp;&nbsp; <input type="checkbox" name="day" value="sat"/>Sat<br/>
    <input type="checkbox" name="day" value="sun"/>Sun<br/><br/>
</div>

And

function getDays(select){
    var selectedString = select.options[select.selectedIndex].value;
    if(selectedString == 4)
    {
        document.getElementById("days_target").style.display = "block";
    }else {
        document.getElementById("days_target").style.display = "none";
    }
}

Demo: fiddle

You can simplify by passing the value of the select:

html:

<b>Please enter days required</b><br>
        <select name="days" id="days" style="width:200px;" onchange="getDays(this.value)">
            <option value="0">Select</option>
            <option value="1">Mon-Fri</option>
            <option value="2">Mon-Fri</option>
            <option value="3">Mon-Fri</option>
            <option value="4">Bespoke Days</option>
        </select><br><br>
        <div id="days_target" style="display:none;">
            <input type="checkbox" name="day" value="mon">Mon &nbsp;&nbsp; <input type="checkbox" name="day" value="tue">Tue<br>
            <input type="checkbox" name="day" value="wed">Wed &nbsp;&nbsp; <input type="checkbox" name="day" value="thr">Thr<br>
            <input type="checkbox" name="day" value="fri">Fri &nbsp;&nbsp; <input type="checkbox" name="day" value="sat">Sat<br>
            <input type="checkbox" name="day" value="sun">Sun<br><br>
        </div>

js:

function getDays(value){
        if(value == 4)
        {
            document.getElementById("days_target").style.display = "block";
        }
        else
        {
            document.getElementById("days_target").style.display = "none";
        }
    }