What's the best way to realize it?To do a check on blur ,or on key up/down ?
I'd use the jQuery Validation framework...it'll handle it all for you.
It's highly customisable too.
Ok, if you don't want to use plugins and you want to make your life harder ;) then you have to see when you want the error to be displayed. Jquery Validation uses events on submit, blur and keydown. So you will need to attach all these events and make sure everything is valid on submit. Then let's say you were looking for a valid email, which is not valid. So on keydown you can check to see if the email is valid and remove the warning. All this work has been done in the plugin.