By default you must export the component as a whole by creating a solution (managed or unmanaged) and then selecting "add existing component".
You can export portions of a component but to do that you must delve in to the XML of the exported solution. For example you can export just the ribbon for an entity, or just the form of an entity. The method is to just edit the customizations file and remove the portions you don't want to export/update. Your initial logic might be that if it is removed from the customizations file then won't it mean that it will be removed from the destination organization on import? No, that's not how it works. Read on for more info.
If you want to delete a field from an entity for example within a managed solution, how would you do that? You can't delete it from an org, export the updated version of the solution and import it in to the destination org as CRM will just keep the field. You must create a temporary solution with the entity with the attribute removed. Import the temporary solution to the destination org so that you can remove the actual solution you want to update. After you remove the actual solution you want to update, you update it to not have the field then re import. Once reimported you can remove the temporary solution. Messy.
Hope that helps.