var date = new Date();

alert( date.getFullYear() + ("0" + (date.getMonth() + 1)).slice(-2) + ("0" + date.getDate()).slice(-2) + ("0" + date.getHours() ).slice(-2) + ("0" + date.getMinutes()).slice(-2) + ("0" + date.getSeconds()).slice(-2) );

edit

function pad2(n) { return n < 10 ? '0' + n : n }

var date = new Date();
    
alert( date.getFullYear().toString() + pad2(date.getMonth() + 1) + pad2( date.getDate()) + pad2( date.getHours() ) + pad2( date.getMinutes() ) + pad2( date.getSeconds() ) );

From ES6 onwards you can use template strings to make it a little shorter:

var now = new Date();
var todayString = `${now.getFullYear()}-${now.getMonth()}-${now.getDate()}`;

This solution does not zero pad. Look to the other good answers to see how to do that.

Timestamp in milliseconds

To get the number of milliseconds since Unix epoch, call Date.now:

Date.now()

Alternatively, use the unary operator + to call Date.prototype.valueOf:

+ new Date()

Alternatively, call valueOf directly:

new Date().valueOf()

To support IE8 and earlier (see compatibility table), create a shim for Date.now:

if (!Date.now) {
    Date.now = function() { return new Date().getTime(); }
}

Alternatively, call getTime directly:

new Date().getTime()

Timestamp in seconds

To get the number of seconds since Unix epoch, i.e. Unix timestamp:

Math.floor(Date.now() / 1000)

Alternatively, using bitwise-or to floor is slightly faster, but also less readable and may break in the future (see explanations 1, 2):

Date.now() / 1000 | 0

Timestamp in milliseconds (higher resolution)

Use performance.now:

var isPerformanceSupported = (
    window.performance &&
    window.performance.now &&
    window.performance.timing &&
    window.performance.timing.navigationStart
);

var timeStampInMs = (
    isPerformanceSupported ?
    window.performance.now() +
    window.performance.timing.navigationStart :
    Date.now()
);

console.log(timeStampInMs, Date.now());