<form id="form">
    <input type="text" name="aaa">
    <select name="bbb">
        <option value="">请选择</option>
        <option value="1">选项1</option>
        <option value="2">选项2</option>
        <option value="3">选项3</option>
    </select>
    <button type="button" id="submit">提交</button>
</form>

//提交表单
$('#submit').click(function(){
    var data = {};
    var t = $('#form').serializeArray();
    $.each(t, function() {
        data [this.name] = this.value;
    });
    console.log(data);
    if(data.aaa == ''){alert("aaa不能为空！");return false;}
    if(data.bbb == ''){alert("bbb不能为空！");return false;}
    $('#form').submit();
});

HTML5 required属性：

<input type="text" name="aaa" required="required">
<select name="bbb" required="required">
    <option value="">请选择</option>
    <option value="1">选项1</option>
    <option value="2">选项2</option>
    <option value="3">选项3</option>
</select>
    
<button type="submit">提交</button>