Edit the sidebar.php file - either with a text editor or with the wordpress editor - scroll down until you see the categories listing. Look for optioncount=1, and change it to optioncount=0. It should look something like this - <?php wp_list_cats('optioncount=0&hierarchical=1'); ?> But some of the sidebar codes I have seen looks different from that. Maybe that will get you pointed in the right direction.
I don't think that is what he is talking about. When you go to your categories in your dashboard, hover your mouse over each category, and look at the bottom right of your browser, and it will show the url of the category with the corresponding number.
Some themes show the number of articles next to the category name in the sidebar. So that is you have 10 articles posted in category "abc" - in the sidebar the number 10 will be next to ABC category link. The theme I am using right now, that was how it was. The number of articles was in the sidebar. Changing that setting to 0 turned the numbers off. But I guess that is not what he was asking about.
Dashboard->Appearance->Widgets->Categories Uncheck "Show post counts" Otherwise, see if they go away with <?php wp_list_categories(); ?>
yes "Cash Nebula" is right, you can turn off removing the "show_count=1" from wp_list_categories('........&show_count=1')