onmouseover="document.getElementsByClassName().style.background='color'"

document.getElementsByClassName()
                   ^_______

<th id="colorswitcher A" onmouseover="document.getElementsByClassName('a').style.background='red'">a</th>
<th id="colorswitcher B" onmouseover="document.getElementsByClassName('a').style.background='blue'">b</th>

window.onload=function(){
    var aColl = document.getElementsByClassName('a'); //Cache the collection here, so that even a new element added with the same class later we can avoid qurying this again by using the cached collection.
    var bColl = document.getElementsByClassName('b');

    document.getElementById('A').addEventListener('mouseover', function(){
        changeColor(aColl, 'red');
    });

    document.getElementById('B').addEventListener('mouseover', function(){
        changeColor(bColl, 'blue');
    });
}
function changeColor(coll, color){

    for(var i=0, len=coll.length; i<len; i++)
    {
        coll[i].style["background-color"] = color;
    }
}