Simple widget causing whitescreen

register_widget( 'latest_mag_issues_widget' );

should be

register_widget( 'latest_mag_issue_widget' );

Update
After fixing that though, there are a couple of other errors. Here’s the complete updated copy of your code that I could get working:

class latest_mag_issue_widget extends WP_Widget {

    function latest_mag_issue_widget() {
        $widget_ops = array( 'classname' => 'widget_latest_issues', 'description' => 'Latest Mag Issues Widget for Media24 Business & Customs' );
        $this->WP_Widget( 'latest_issues', 'Latest Issues', $widget_ops );
    } //end constructor

    function widget( $args, $instance ) {
        extract( $args );

        // widget options
        $title = apply_filters( 'widget_title', $instance['title'] );
        $pastissues = $instance['pastissues'];
        $thisissue = $instance['thisissue'];

        echo $before_widget;

        echo '<div class="widget widget-latest-issue">';

        if ( $title ) {
            echo $before_title . $title . $after_title;
        }

        $wp_query = new WP_Query();
        $wp_query->query( array( 'post_type' => 'mag-issue', 'post_status' => 'publish', 'paged' => 1, 'posts_per_page' => 1 ) );

        if ( have_posts() ) : while ( have_posts() ) : the_post();
            $image_url = get_post_meta( $post->ID, 'press_smart', true );
            $image_title = the_title();
            $image_thumbnail = the_post_thumbnail( array(125,166) );

            echo '
                <div id="latest-issue">
                    <h3>Latest Issue</h3>
                    <a class="fl" href="' . $image_url . '" title="' . $image_title .'">' . $image_thumbnail . '</a> <br/>
                </div>
            ';

            if ( $thisissue == true ) {
                echo '
                    <div id="this-issue">
                        <h3>In This Issue</h3>
                        <ul>';

                        for ( $i = 1; $i < 10; $i++ ) {
                            $headline_meta = get_post_meta( $post->ID , 'headline_' . $i  , true );
                            if ( $headline_meta ) { echo '<li>' . $headline_meta . '</li><hr>';}
                        }

                echo '
                        </ul>
                    </div>
                ';
            }

            if ( $pastissues == true ) {

                $wp_query->query(array( 'post_type' => 'mag-issue', 'post_status' => 'publish', 'paged' => 1, 'posts_per_page' => 9, 'offset' => 1 ));

                if ( have_posts() ) : while ( have_posts() ) : the_post();

                    $post_url = get_post_meta( $post->ID, 'press_smart', true );
                    $post_title = the_title();

                    echo '
                    <div id="past-issues">
                        <h3>Past Issues</h3>
                        <ul>
                            <li>
                                <a href="' . $post_url  . '">' . $post_title . '</a>
                            </li>';
                    echo '
                        </ul>
                    </div>
                    ';

                    endwhile; endif; wp_reset_postdata();
            }
        echo '</div>';

        echo $after_widget;

        endwhile; endif; wp_reset_query();
    } //end function widget()

    function update( $new_instance, $old_instance ) {
        $instance = $old_instance;
        $instance['title'] = strip_tags( $new_instance['title'] );
        $instance['thisissue'] = strip_tags( $new_instance['thisissue'] );
        $instance['pastissues'] = strip_tags( $new_instance['pastissues'] );
        return $instance;
    } //end function update()


    function form($instance) {

        $title = esc_attr( $instance['title'] );
        $thisissue = esc_attr( $instance['thisissue'] );
        $pastissues = esc_attr( $instance['pastissues'] );

        ?>

        <p>
            <label for="<?php echo $this->get_field_id('title'); ?>"><?php _e('Widget Title'); ?></label>
            <input class="widefat" id="<?php echo $this->get_field_id('title'); ?>" name="<?php echo $this->get_field_name('title'); ?>" type="text" value="<?php echo $title; ?>" />
        </p>

        <p>
            <input id="<?php echo $this->get_field_id('thisissue'); ?>" name="<?php echo $this->get_field_name('thisissue'); ?>" type="checkbox" value="1" <?php checked( '1', $thisissue ); ?>/>
            <label for="<?php echo $this->get_field_id('thisissue'); ?>"><?php _e('Display headlines?'); ?></label>
        </p>

        <p>
            <input id="<?php echo $this->get_field_id('pastissues'); ?>" name="<?php echo $this->get_field_name('pastissues'); ?>" type="checkbox" value="1" <?php checked( '1', $pastissues ); ?>/>
            <label for="<?php echo $this->get_field_id('pastissues'); ?>"><?php _e('Display past issues?'); ?></label>
        </p>

        <?php
    } //end function form()

}

// Register the widget
add_action( 'widgets_init', 'register_mag_issues_widget' );
function register_mag_issues_widget() {
    register_widget( 'latest_mag_issue_widget' );
}