How do I know which version of Javascript I'm using?

Click on this link to see which version your BROWSER is using: http://jsfiddle.net/Ac6CT/

You should be able filter by using script tags to each JS version.

<script type="text/javascript">
  var jsver = 1.0;
</script>
<script language="Javascript1.1">
  jsver = 1.1;
</script>
<script language="Javascript1.2">
  jsver = 1.2;
</script>
<script language="Javascript1.3">
  jsver = 1.3;
</script>
<script language="Javascript1.4">
  jsver = 1.4;
</script>
<script language="Javascript1.5">
  jsver = 1.5;
</script>
<script language="Javascript1.6">
  jsver = 1.6;
</script>
<script language="Javascript1.7">
  jsver = 1.7;
</script>
<script language="Javascript1.8">
  jsver = 1.8;
</script>
<script language="Javascript1.9">
  jsver = 1.9;
</script>

<script type="text/javascript">
  alert(jsver);
</script>

My Chrome reports 1.7

Blatantly stolen from: http://javascript.about.com/library/bljver.htm

Rather than finding which version you are using you can rephrase your question to "which version of ECMA script does my browser's JavaScript/JSscript engine conform to".

For IE :

alert(@_jscript_version);      //IE

Refer Squeegy's answer for non-IE versions :)

All of todays browsers use at least version 1.5:
http://en.wikipedia.org/wiki/ECMAScript#Dialect

Concerning your tutorial site, the information there seems to be extremely outdated, I beg you to head over to MDC and read their Guide:
https://developer.mozilla.org/en/JavaScript/Guide

You may still want to watch out for features which require version 1.6 or above, as this might give Internet Explorer some troubles.