Change text and styling of a form control when a slider value changes
You are duplicating a lot of code here. You said you wanted an array, so make an array containing all the values that change and then use the slideEvt.value to recall the values from the array at the correct index..

Something like this maybe.. no if() statements required (at least not for the slideEvt.value):

$("input#communication").on("slide", function(slideEvt) {
    var slideEvtVals = [
      {color: '#D0011B', css: {'border': '5px solid #fff', 'box-shadow': '0 0 0 2px #D0011B'}, text: 'Amir is a terrible communicator.'}
      {color: '#D0011B', text: 'Amir struggles, needs a lot of help to improve communication skills.'}
      {color: '#D0011B', css: {'border': '5px solid #fff', 'box-shadow': '0 0 0 2px #D0011B'}, text: 'Amir needs some help to improve communication skills.', handleBG: '#D0011B', selBg: '#D0011B'}
      {color: '#F47621', css: {'border': '5px solid #fff', 'box-shadow': '0 0 0 2px #F47621'}, text: 'Amir has below average communication skills.', handleBG: '#F47621', selBg: '#F47621'}
      {color: '#F47621', text: 'Amir has adequate communications skills.'}
    ];

    var index = slideEvt.value-1;
    $('#slider-val'+slideEvt.value).css('color', slideEvtVals[index].color);
    $('.slider-label').not('#slider-val'+slideEvt.value).css('color', '#ccc');
    if(slideEvtVals[index].css) $('.slider-handle').css(slideEvtVals[index].css);
    $("#desc").text(slideEvtVals[index].text);
    $("#desc").text(slideEvtVals[index].text);
    if(slideEvtVals[index].handleBG) $('.slider-handle').css('background', slideEvtVals[index].handleBG);
    if(slideEvtVals[index].selBg) $('.slider-selection').css('background', slideEvtVals[index].selBg);
});