Shortcode functions accept a second parameter which contains the value in between the shortcode opening and closing tags:
function youtube( $atts, $value="http://" ) {
extract( shortcode_atts( array(
"width" => '620',
"height" => '350',
"name"=> 'movie',
"allowFullScreen" => 'true',
"allowScriptAccess"=>'always',
), $atts ) );
return '<object style="height: '.$height.'px; width: '.$width.'px"><param name="'.$name.'" value="'.$value.'"><param name="allowFullScreen" value="'.$allowFullScreen.'"></param><param name="allowScriptAccess" value="'.$allowScriptAccess.'"></param><embed src="'.$value.'" type="application/x-shockwave-flash" allowfullscreen="'.$allowFullScreen.'" allowScriptAccess="'.$allowScriptAccess.'" width="'.$width.'" height="'.$height.'"></embed></object>';
}
add_shortcode( 'yt', 'youtube' );