Remove site name from tag

The best (and easiest) thing to do is to use the wp_title filter.

First, clean up your call to <?php wp_title(); ?> in your template. Replace what you have with this:

 wp_title( '|', true, 'right' );

Then, in functions.php (or in a child Theme functions.php; normal caveats apply), add the following:

function wpse95147_filter_wp_title( $title ) {
    if ( is_single() || ( is_home() && !is_front_page() ) || ( is_page() && !is_front_page() ) ) {
        $title = single_post_title( '', false );
    }
    return $title;
}
add_filter( 'wp_title', 'wpse95147_filter_wp_title' );

This filter will completely overwrite the $title output in single-page contexts. (The actual conditional is taken directly from core, from the wp_title() function definition itself.)

Last I checked, WordPress doesn’t actually output the site title anywhere in wp_title(); so if you’re seeing that, something is adding it (perhaps your Theme?). Nevertheless, if you want to output it, just write a complimentary conditional in the above filter; e.g.:

function wpse95147_filter_wp_title( $title ) {
    if ( is_single() || ( is_home() && !is_front_page() ) || ( is_page() && !is_front_page() ) ) {
        $title = single_post_title( '', false );
    }
    if ( is_front_page() && ! is_page() ) {
        $title = esc_attr( get_bloginfo( 'name' ) );
    }
    return $title;
}
add_filter( 'wp_title', 'wpse95147_filter_wp_title' );