How can you call a controller method from JavaScript in salesforce?

Hello,

using the tag apex:actionFunction(A component that provides support for invoking controller action methods directly from JavaScript code using an AJAX request. An component must be a child of an component.)

for an exaple :-

Controller:-

public class controller{
public pageReference noiseSearch(){
//logic
return null;
}
}

vf page:-
So your button will be

<apex:commandButton value="Search" onclick="check();" reRender="nPbt" />

And your script function

function check(){
var user = document.getElementById('pg:frm:nPB:ip').value;
if(user == ''){
alert('Please Enter noise Name before Proceeding.');
return true;
}
else
noiseSearchAF();
return true;
}