04/2020: Corrected old answer
Use :selected pseudo selector on the selected options and then use the .val function to get the value of the option.
$('select[name=selector] option').filter(':selected').val()
Side note: Using filter is better then using :selected selector directly in the first query.
If inside a change handler, you could use simply this.value to get the selected option value. See demo for more options.
//ways to retrieve selected option and text outside handler
console.log('Selected option value ' + $('select option').filter(':selected').val());
console.log('Selected option value ' + $('select option').filter(':selected').text());
$('select').on('change', function () {
//ways to retrieve selected option and text outside handler
console.log('Changed option value ' + this.value);
console.log('Changed option text ' + $(this).find('option').filter(':selected').text());
});<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/1.12.4/jquery.min.js"></script>
<select>
<option value="1" selected>1 - Text</option>
<option value="2">2 - Text</option>
<option value="3">3 - Text</option>
<option value="4">4 - Text</option>
</select>Old answer:
Edit: As many pointed out, this does not returns the selected option value.
~~Use .val to get the value of the selected option. See below,
$('select[name=selector]').val()~~