When to use async false and async true in ajax function in jquery

It's not relative to performance...

You set async to false, when you need that ajax request to be completed before the browser passes to other codes:

<script>
    // ...
    $.ajax(... async: false ...); // Hey browser! first complete this request, 
                                  // then go for other codes

    $.ajax(...); // Executed after the completion of the previous async:false request.
</script>

1. When async setting is set to false, a Synchronous call is made instead of an Asynchronous call.
2. When the async setting of the jQuery AJAX function is set to true then a jQuery Asynchronous call is made. AJAX itself means Asynchronous JavaScript and XML and hence if you make it Synchronous by setting async setting to false, it will no longer be an AJAX call.
3. for more information please refer this link

It is best practice to go asynchronous if you can do several things in parallel (no inter-dependencies). If you need it to complete to continue loading the next thing you could use synchronous, but note that this option is deprecated to avoid abuse of sync:

jQuery.ajax() method's async option deprecated, what now?