RemoteController.cls

global class RemoteController {

    @RemoteAction
    global static String test0Str(){
        return '0 arguments';
    }
    
    @RemoteAction
    global static String test1Str(String str){
        return str;
    }
    
    @RemoteAction
    global static String test2Str(String str1,String str2){
        return str1+ ' ' + str2;
    }
}

RemoteController.page

<apex:page controller="RemoteController"  >

<script> 
function handleReturn(result, event) {
    if (event.status) {
    document.getElementById("output").innerHTML = result + ', ' + document.getElementById("output").innerHTML;
    } else if (event.type === 'exception') {
    document.getElementById("output").innerHTML = event.message;
    } else {
    document.getElementById("output").innerHTML = event.message;
    }
}
</script>
<div id="output" />

<form >
<div>
    <input value="summer12 0 params" onclick="Visualforce.remoting.Manager.invokeAction('{!$RemoteAction.RemoteController.test0Str}', handleReturn,{escape: true})" type="button" />
    <input value="spring12 0 params" onclick="RemoteController.test0Str( handleReturn,{escape: true})" type="button" />
</div>
<div>
    <input value="summer12 1 params" onclick="Visualforce.remoting.Manager.invokeAction('{!$RemoteAction.RemoteController.test1Str}','1 argument', handleReturn,{escape: true})" type="button" />
    <input value="spring12 1 params" onclick="RemoteController.test1Str('1 argument', handleReturn,{escape: true})" type="button" />
</div>
<div>
    <input value="summer12 2 params" onclick="Visualforce.remoting.Manager.invokeAction('{!$RemoteAction.RemoteController.test2Str}','2','arguments', handleReturn,{escape: true})" type="button" />    
    <input value="spring12 2 params" onclick="RemoteController.test2Str('2','arguments',  handleReturn,{escape: true})" type="button" />
</div>
</form>
</apex:page>