You should use a library like jQuery UI Autocomplete or Twitter Bootstrap Typeahead on the client side.
On the server side you create a method to handle your AJAX call and provide a JSON or XML response to the used JS library (whichever you prefer).
However if your values are predefined and the list is small enough, you may but the values already in the javascript as your datasource to avoid requests to the server.
If you are using a database to select your suggestions, you may use something like this:
SELECT id, value FROM your_table WHERE value LIKE `%:keyword%`;
And replace :keyword
with the provided user input. I would also add an index on the value
field to execute this query faster.