One basic trick you can use for this that I haven't seen fully mentioned. If you want to do an ajax action, or some other work on Enter but don't want to actually submit a form you can do this:
<form onsubmit="Search();" action="javascript:void(0);">
<input type="text" id="searchCriteria" placeholder="Search Criteria"/>
<input type="button" onclick="Search();" value="Search" id="searchBtn"/>
</form>
Setting action="javascript:void(0);" like this is a shortcut for preventing default behavior essentially. In this case a method is called whether you hit enter or click the button and an ajax call is made to load some data.