<select id="countries-select"></select>
<select id="cities-select"></select>

<script>
    var countriesSelect = document.querySelector('#countries-select');
    var citiesSelect = document.querySelector('#cities-select');

    var data = {
        Россия: ['Москва', 'Питер', 'Воронеж'],
        Беларусь: ['Минск', 'Витебск', 'Орша'],
        Украина: ['Киев', 'Львов', 'Одесса'],
    }

    var countries = Object.keys(data);
    addOptions(countriesSelect, countries);

    var defaultCities = data[countries[0]];
    addOptions(citiesSelect, defaultCities);

    countriesSelect.addEventListener('change', function() {
        var cities = data[this.value];
        citiesSelect.length = 0;

        addOptions(citiesSelect, cities);
    });

    function addOptions(select, arr) {
        for (var i = 0; i < arr.length; i++) {
            select.add(new Option(arr[i]));
        }
    }