TommyDBrown,
<!DOCTYPE html>
<html>
<head>
<title>Untitled</title>
<meta charset="utf-8">
<style type="text/css">
.contacts__item--active {
background-color: #D3D3D3;
}
</style>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.12.4/jquery.min.js"></script>
<script>
$(function() {
var city = $(".header__city-item"), contact = $(".contacts__item");
city.on("click", function(event) {
event.preventDefault();
var i = city.index(this);
contact.removeClass("contacts__item--active").eq(i).addClass("contacts__item--active")
})
});
</script>
</head>
<body>
<div class="header__center-inner">
<div class="header__city">
<span>Ваш город: <a class="header__city-link" href="#">выберите город</a></span>
<ul class="header__city-list">
<li class="header__city-item header__city-item--belgorod"><a href="#">Белгород</a></li>
<li class="header__city-item header__city-item--vladimir"><a href="#">Владимир</a></li>
<li class="header__city-item header__city-item--jeleznogorsk"><a href="#">Железногорск</a></li>
<li class="header__city-item header__city-item--kursk"><a href="#">Курск</a></li>
<li class="header__city-item header__city-item--orel"><a href="#">Орёл</a></li>
</ul>
</div>
<ul>
<li class="contacts__item contacts__item--belgorod">Белгород</li>
<li class="contacts__item contacts__item--vladimir">Владимир</li>
<li class="contacts__item contacts__item--jeleznogorsk">Железногорск</li>
<li class="contacts__item contacts__item--kursk">Курск</li>
<li class="contacts__item contacts__item--orel">Орёл</li>
</ul>
</div>
</body>
</html>