How to filter a list with JavaScript? SharePoint 2013

If this is one webpage then you can

• Add an onchange eventhandler on the drop down <select>
• Then use a JS standard document.querySelector to find the matching elements in the list
• You then loop all elements, find the parent TR row, and hide

Something like: disclaimer: didn't not test this code, you are on your own

document.getElementById('mySelect').addEventListener('onchange', function () {
    var employeeName = this.value;
    var matches = document.querySelector("a[href*=" + employeeName + "']");
    matches.forEach(function (match) {
        var row = GetAncestor(match, 'TR');
        row.style.display='none';
    });
});

• GetAncestor is provided by SharePoint
• You have to add ...display='block' yourself to show rows again
• forEach might need a Polyfill in older IE versions

Sharepoint rest api example if Employee is a lookup field and it displays the Title

var url= "/_api/Web/Lists/GetByTitle('childribbon')/Items?"+
           "$select=ID,Title,Employee/Title,Skills"+
           "&$expand=Employee&$filter=Employee/Title eq 'Nderon Hiseni'";
$.ajax({
    url: _spPageContextInfo.webAbsoluteUrl + url,
    type: "GET",
    headers: {
        "accept": "application/json;odata=verbose",
    },
    success: function (data) {
        console.log(data.d.results);
    },
    error: function (error) {
        alert(JSON.stringify(error));
    }
});

Otherwise, if Employee is not a lookup field then the url should be

var url= "/_api/Web/Lists/GetByTitle('childribbon')/Items?"+
           "$select=ID,Title,Title,Skills"+
           "&$filter=Employee eq 'Nderon Hiseni'";

You can read this article for better understanding http://www.codeproject.com/Articles/990131/CRUD-Operation-to-List-Using-SharePoint-Rest-API