Correct Answer:

function addDays(date, days) {
  var result = new Date(date);
  result.setDate(result.getDate() + days);
  return result;
}

Incorrect Answer:

This answer sometimes provides the correct result but very often returns the wrong year and month. The only time this answer works is when the date that you are adding days to happens to have the current year and month.

// Don't do it this way!
function addDaysWRONG(date, days) {
  var result = new Date(); // not instatiated with date!!! DANGER
  result.setDate(date.getDate() + days);
  return result;
}

Proof / Example

Check this JsFiddle

// Correct
function addDays(date, days) {
    var result = new Date(date);
    result.setDate(result.getDate() + days);
    return result;
}

// Bad Year/Month
function addDaysWRONG(date, days) {
    var result = new Date();
    result.setDate(date.getDate() + days);
    return result;
}

// Bad during DST
function addDaysDstFail(date, days) {
    var dayms = (days * 24 * 60 * 60 * 1000);
    return new Date(date.getTime() + dayms);    
}

// TEST
function formatDate(date) {
    return (date.getMonth() + 1) + '/' + date.getDate() + '/' + date.getFullYear();
}

$('tbody tr td:first-child').each(function () {
    var (this);
    var ('<td/>').insertAfter($in).addClass("answer");
    var $outFail = $('<td/>').insertAfter(outDstFail = $('<td/>').insertAfter($outFail);
    var date = new Date($in.text());
    var correctDate = formatDate(addDays(date, 1));
    var failDate = formatDate(addDaysWRONG(date, 1));
    var failDstDate = formatDate(addDaysDstFail(date, 1));

    $out.text(correctDate);
    $outFail.text(failDate);
    $outDstFail.text(failDstDate);
    $outFail.addClass(correctDate == failDate ? "right" : "wrong");
    $outDstFail.addClass(correctDate == failDstDate ? "right" : "wrong");
});

body {
    font-size: 14px;
}

table {
    border-collapse:collapse;
}
table, td, th {
    border:1px solid black;
}
td {
    padding: 2px;
}

.wrong {
    color: red;
}
.right {
    color: green;
}
.answer {
    font-weight: bold;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<table>
    <tbody>
        <tr>
            <th colspan="4">DST Dates</th>
        </tr>
        <tr>
            <th>Input</th>
            <th>+1 Day</th>
            <th>+1 Day Fail</th>
            <th>+1 Day DST Fail</th>
        </tr>
        <tr><td>03/10/2013</td></tr>
        <tr><td>11/03/2013</td></tr>
        <tr><td>03/09/2014</td></tr>
        <tr><td>11/02/2014</td></tr>
        <tr><td>03/08/2015</td></tr>
        <tr><td>11/01/2015</td></tr>
        <tr>
            <th colspan="4">2013</th>
        </tr>
        <tr>
            <th>Input</th>
            <th>+1 Day</th>
            <th>+1 Day Fail</th>
            <th>+1 Day DST Fail</th>
        </tr>
        <tr><td>01/01/2013</td></tr>
        <tr><td>02/01/2013</td></tr>
        <tr><td>03/01/2013</td></tr>
        <tr><td>04/01/2013</td></tr>
        <tr><td>05/01/2013</td></tr>
        <tr><td>06/01/2013</td></tr>
        <tr><td>07/01/2013</td></tr>
        <tr><td>08/01/2013</td></tr>
        <tr><td>09/01/2013</td></tr>
        <tr><td>10/01/2013</td></tr>
        <tr><td>11/01/2013</td></tr>
        <tr><td>12/01/2013</td></tr>
        <tr>
            <th colspan="4">2014</th>
        </tr>
        <tr>
            <th>Input</th>
            <th>+1 Day</th>
            <th>+1 Day Fail</th>
            <th>+1 Day DST Fail</th>
        </tr>
        <tr><td>01/01/2014</td></tr>
        <tr><td>02/01/2014</td></tr>
        <tr><td>03/01/2014</td></tr>
        <tr><td>04/01/2014</td></tr>
        <tr><td>05/01/2014</td></tr>
        <tr><td>06/01/2014</td></tr>
        <tr><td>07/01/2014</td></tr>
        <tr><td>08/01/2014</td></tr>
        <tr><td>09/01/2014</td></tr>
        <tr><td>10/01/2014</td></tr>
        <tr><td>11/01/2014</td></tr>
        <tr><td>12/01/2014</td></tr>
        <tr>
            <th colspan="4">2015</th>
        </tr>
        <tr>
            <th>Input</th>
            <th>+1 Day</th>
            <th>+1 Day Fail</th>
            <th>+1 Day DST Fail</th>
        </tr>
        <tr><td>01/01/2015</td></tr>
        <tr><td>02/01/2015</td></tr>
        <tr><td>03/01/2015</td></tr>
        <tr><td>04/01/2015</td></tr>
        <tr><td>05/01/2015</td></tr>
        <tr><td>06/01/2015</td></tr>
        <tr><td>07/01/2015</td></tr>
        <tr><td>08/01/2015</td></tr>
        <tr><td>09/01/2015</td></tr>
        <tr><td>10/01/2015</td></tr>
        <tr><td>11/01/2015</td></tr>
        <tr><td>12/01/2015</td></tr>
    </tbody>
</table>

Notice that Eastern Daylight Time is -4 hours and that the hours on the date you're getting back are 20.

20h + 4h = 24h

which is midnight of 2011-09-24. The date was parsed in UTC (GMT) because you provided a date-only string without any time zone indicator. If you had given a date/time string w/o an indicator instead (new Date("2011-09-24T00:00:00")), it would have been parsed in your local timezone. (Historically there have been inconsistencies there, not least because the spec changed more than once, but modern browsers should be okay; or you can always include a timezone indicator.)

You're getting the right date, you just never specified the correct time zone.

If you need to access the date values, you can use getUTCDate() or any of the other getUTC*() functions:

var d,
  days;
d = new Date('2011-09-24');
days = ['Sun', 'Mon', 'Tues', 'Wed', 'Thurs', 'Fri', 'Sat'];
console.log(days[d.getUTCDay()]);

You could use the Solution from UTC Date Conversion. Which basicall does the following:

console.log(new Date("2014-12-23"));
console.log(convertDateToUTC(new Date("2014-12-23")));

function convertDateToUTC(date) { 
return new Date(date.getUTCFullYear(), date.getUTCMonth(), date.getUTCDate(), date.getUTCHours(), date.getUTCMinutes(), date.getUTCSeconds()); 
}

The output would be like that in the console (for me at least :D)

Tue Dec 23 2014 01:00:00 GMT+0100 (Mitteleuropäische Zeit)
Tue Dec 23 2014 00:00:00 GMT+0100 (Mitteleuropäische Zeit)

What time zone does the Javascript new Date() use?

Date objects store the number of milliseconds since The Epoch (Jan 1st 1970 at midnight GMT). They have methods like getDay and getMonth and such that provide values in the local timezone, and also functions like getUTCDay and getUTCMonth that provide values in UTC (loosely, GMT).

If you're parsing a string, you need to be sure that the string is in a format that Date knows how to parse. The only defined format in the specification is a simplified derivative of ISO-8601: YYYY-MM-DDTHH:mm:ss.sssZ. But that was only added in ES5 (and then updated in ES2015 and ES2016). The time zone of the string is determined by whether it has a UTC offset indicator (Z, -08:00, +05:30, etc.) and whether it's date-only or date/time. Here are some examples:

function test(str) {
    const dt = new Date(str);
    const p = document.createElement("pre");
    p.textContent =
        `String: "${str}"\n` +
        `UTC:    ${dt.toISOString()}\n` +
        `Local:  ${dt.toLocaleString()}`
    ;
    document.body.appendChild(p);
}

// No UTC offset provided, date-only form => parsed as UTC:
test("2015-08-09");

// No UTC offset provided, date/time form => parsed as local time:
test("2015-08-09T09:32:54");

// Has UTC offset "Z" (for "no offset") => parsed as UTC:
test("2015-08-09T09:32:54.427Z");

// Has UTC offset -08:00 => parsed as UTC minus eight hours:
test("2015-08-09T09:32:54.427-08:00");

I don't recommend relying on it, but even though it's not specified, all major JavaScript engines will successfully parse the American-specific formats MM/DD/YYYY, MM/DD/YYYY hh:mm, MM/DD/YYYY hh:mm:ss, or MM/DD/YYYY hh:mm:ss.SSS (although the milliseconds portion is ignored by some). Note the order in the date portion: month, day, year. Having a UTC offset indicator is not broadly supported and will cause an error on most engines, so don't include one. Without one, all of those are parsed as local time (even the date-only one). Note that the date field separator must be / (08/09/2015), not - (08-09-2015).

function test(str) {
    const dt = new Date(str);
    const p = document.createElement("pre");
    p.textContent =
        `String: "${str}"\n` +
        `UTC:    ${dt.toISOString()}\n` +
        `Local:  ${dt.toLocaleString()}`
    ;
    document.body.appendChild(p);
}

test("08/09/2015");
test("08/09/2015 09:32");
test("08/09/2015 09:32:54");
test("08/09/2015 09:32:54.427");

But again: I don't recommend relying on that.

new Date creates a new Date object that you can modify or initialize with a different date while Date returns a string of the current date/time, ignoring its arguments.