Update

As pointed in the comments since EPiServer 8 there is a simpler way to define tab ordering. Just declare a static class with constant fields which will be the names of the tabs, add GroupDefinitions attribute to the class and use DisplayAttribute's Order property to define an index of the tab. My previous example now looks like this:

[GroupDefinitions]
public static class TabNames
{
    [Display(Order = 100)]
    public const string Sidebar = "Sidebar";

    [Display(Order = 110)]
    public const string Description = "Description";
}

Now use those in the content type:

[ContentType(GUID = "a35c0203-e548-4918-b932-05205cc8c491", Order = 1)]
public class StandardPage : EditorialPageBase
{
    [Display(Name = "Text", Order = 10, GroupName = TabNames.Description)]
    public virtual string Text { get; set; }

    [Display(Name = "Sidebar", Order = 10, GroupName = TabNames.Sidebar)]
    public virtual ContentArea SidebarContentArea { get; set; }
}

And here is the result:

Then try to change indexes of the tab names:

[GroupDefinitions]
public static class TabNames
{
    [Display(Order = 110)]
    public const string Sidebar = "Sidebar";

    [Display(Order = 100)]
    public const string Description = "Description";
}

And tab order gets changed:

It is also possible to set access rights for the tab. In the example below I am setting administer access right on Sidebar tab.

[GroupDefinitions]
public static class TabNames
{
    [RequiredAccess(AccessLevel.Administer)]
    [Display(Order = 100)]
    public const string Sidebar = "Sidebar";

    [Display(Order = 110)]
    public const string Description = "Description";
}

The documentation mentions that it is also possible to redefine order of system tabs but when I tried to redefine Content tab in my Commerce project I got an exception. Here is the code I used to redefine it:

[GroupDefinitions]
public static class TabNames
{
    [Display(Order = 1000)]
    public const string Content = SystemTabNames.Content;
}

And here is an exception:

The 'Information' has been defined more than once, in the 'EPiServer.Commerce.Catalog.DataAnnotations.TabNames' and  in the 'CommerceTest.Web.TabNames'.

For more information about tabs look at EPiServer Documentation.

Original post

First of all, let's define a custom attribute. It has two properties - Name and Index. Name is a tab's name which is used on DisplayAttribute's GroupName parameter. Index sets tab's sort index.

[AttributeUsage(AttributeTargets.Class, AllowMultiple = true)]
public class CustomTabAttribute : Attribute
{
    public CustomTabAttribute(string name, int index)
    {
        if (name == null) throw new ArgumentNullException("name");
        Name = name;
        Index = index;
    }

    public string Name { get; private set; }
    public int Index { get; private set; }
}

Next step, is registering of this attribute in ITabDefinition repository. So, create a new initializable module and in the Initialize method look up for all content types which uses our new CustomTab. Then add found tabs to the tab definition repository.

public void Initialize(InitializationEngine context)
{
    var tabDefinitionRepository =
                        ServiceLocator.Current.GetInstance<ITabDefinitionRepository>();
    var contentTypeRepo =
                        ServiceLocator.Current.GetInstance<IContentTypeRepository>();

    var customTabs = contentTypeRepo.List()
                        .SelectMany(CustomTabAttributes);

    foreach (var tab in customTabs)
    {
        AddTabToList(tabDefinitionRepository,
            new TabDefinition
            {
                Name = tab.Name,
                RequiredAccess = AccessLevel.Edit,
                SortIndex = tab.Index
            });
    }
}

private IEnumerable<CustomTabAttribute> CustomTabAttributes(ContentType x)
{
    if (x.ModelType == null)
    {
        return Enumerable.Empty<CustomTabAttribute>();
    }

    return Attribute.GetCustomAttributes(x.ModelType, typeof (CustomTabAttribute))
                    .Cast<CustomTabAttribute>();
}

private void AddTabToList(
  ITabDefinitionRepository tabDefinitionRepository,
  TabDefinition definition)
{
    var existingTab = GetExistingTabDefinition(tabDefinitionRepository, definition);

    if (existingTab != null)
    {
        definition.ID = existingTab.ID;
    }

    tabDefinitionRepository.Save(definition);
}

private static TabDefinition GetExistingTabDefinition(
  ITabDefinitionRepository tabDefinitionRepository,
  TabDefinition definition)
{
    return
        tabDefinitionRepository.List()
            .FirstOrDefault(t =>
                  t.Name.Equals(
                        definition.Name,
                        StringComparison.InvariantCultureIgnoreCase));
}

And here is the usage of new CustomTab. Just decorate your content type with CustomTab attribute, give it a name (use constants for it), set index for the tab and add some property to this tab by setting DisplayAttribute's GroupName property with tab's name.

[ContentType(GUID = "a35c0203-e548-4918-b932-05205cc8c491", Order = 1)]
[CustomTab("Sidebar", 100)]
[CustomTab("Description", 110)]
public class StandardPage : EditorialPageBase
{
    [Display(Name = "Text", Order = 10, GroupName = "Description")]
    public virtual string Text { get; set; }

    [Display(Name = "Sidebar", Order = 10, GroupName = "Sidebar")]
    public virtual ContentArea SidebarContentArea { get; set; }
}

In the given example Sidebar tab will be displayed before Description tab.

Now switch indexes of the attributes:

[CustomTab("Sidebar", 110)]
[CustomTab("Description", 100)]

And see that tab order is changed:

For more information about tabs look at EPiServer Documentation.