My requirement is to authenicate the user 2) auto populate the text box with similiar names.For eg, when the user types in 'r',i need to fetch all the names starting with 'r' like 'rita','rosy','roseline' etc and show as a dropdown list.
If Ajax is your main aim two really useful libraries would be Prototype & JQuery Prototype is a good library to begin with - its pretty easier to use.
My personal suggestion would be to go for Jquery. Its a bit harder and I think it definitely needs to be learned (recommended book Jquery in Action - I really loved it) but it well worth the time spend to learn it.
Of course if all you need is do a little Ajax then stick to prototype.