If you had read the specific tags I pointed you to, you would have found this example:
<?php wp_list_cats('sort_column=name&optioncount=1&exclude=10, 15'); ?>
It says, list the categories sorted alphabetically by name (not ID), show the number of the posts within each category (shown as (3) after the category - if you want it), and then exclude categories 10 and 15. This is if you don't want the specific categories.
If you are not excluding any, then use:
<?php wp_list_cats('sort_column=name&optioncount=1'); ?>
If you don't want the list in alphabetical order by name, or sorted by ID, then you have a more complicated problem. If your blog is totally new, this isn't a stressor. If it isn't and filled with tons of old posts all in dozens of categories, cry with me.
Okay, so if it is new, then simply go to your Admin screen MANAGE > Categories and delete them all and start over, entering them in the specific order you want them to appear. Now, there is a trick to this. I can't remember right now, but if you do this, they should be removed from the database and the new ones should start with ID numbers of 1, 2, 3. But it might not happen because the database needs to have the ID numbers of the categories be unique, so it might start numbering them 6, 7, 8, but either way, they will be in a numbered sequence so that category 1, would be 6, category 2 would be 7, and so on.
Does that help?