How to show post counts in "Categories" and "Archives" menus in WordPress
In WordPress the Categories menu shows links to your post categories and the Archives menu shows links to the previous months (each link displays the post published in that month). By default, only the names of the categories and months are displayed in the respective menu; the number of posts in each category and month is not indicated. If you want to show the post count in each category/month, you can do this by editing the options of the widget responsible for the menu.
In this case the respective widgets are Categories and Archives. To edit them log in to the admin panel and go to Appearance menu>Widgets sub-menu. On the page that opens you'll see the two widgets listed in one of the widget areas (a sidebar or the footer). Click on the small arrow that's next to the name of each of the two widgets. A panel with the widget settings will expand. For each of the two widgets there's a checkbox Show post counts. Mark it and click on the Save button. Then refresh the frontend of your site to see the result. The post count will be shown in brackets after the name of the category/month.
For more details on managing widgets read the tutorial on how to manage widgets in WordPress.