How can we edit www.example.com/user page?
The page at http://example.com/user has two purposes:


Showing the login form for the users who are not logged in
Showing the profile page for the users who are logged in


In the first case, there isn't anything that you can edit to change how the login form is rendered. What you can do is implementing hook_form_user_login_alter() in the theme you are using, or in a module, to alter how the login form is rendered. In this case, you can use the #prefix and #suffix properties to inject CSS classes to the form fields, and #attached to load a new CSS file containing the CSS styles you need.
The alternative is associating a theme function to render the login form, through #theme.

In the second case, with the exception of the template file used to render the page, there isn't any file you can edit to change how the user profile page appear. Editing the user-profile.tpl.php template file used by the currently used theme is the easier, and faster way to change how the user profile page appears.
The alternatives are:


Changing the page callback associated with the user/%uid path with hook_menu_alter()
Implementing hook_user_view() or hook_entity_view() to add items to the user profile page
Implementing hook_user_view_alter() or hook_entity_view_alter() to alter what the user profile page contains


See also Gisle Hannemyr's answer; if you just need to change the order in which the fields associated with a user account are rendered, or slightly changing how those fields are rendered, that is the answer for you.