You have a few issues with your code:
<select name="dbType" id=dbType"> should be <select name="dbType" id="dbType">
$('this') should be $(this): there is no need for the quotes inside the paranthesis.
use .val() instead of .value() when you want to retrieve the value of an option
when u initialize "selection" do it with a var in front of it, unless you already have done it at the beggining of the function
try this:
$('#dbType').on('change',function(){
if( $(this).val()==="other"){
$("#otherType").show()
}
else{
$("#otherType").hide()
}
});
UPDATE for use with switch:
$('#dbType').on('change',function(){
var selection = $(this).val();
switch(selection){
case "other":
$("#otherType").show()
break;
default:
$("#otherType").hide()
}
});
UPDATE with links for jQuery and jQuery-UI:
<script src="//ajax.googleapis.com/ajax/libs/jquery/1.9.1/jquery.min.js" ></script>
<script src="//ajax.googleapis.com/ajax/libs/jqueryui/1.10.2/jquery-ui.min.js"></script>??