WPDB: how to get the value of a field in a custom database table

OK, so first of all, you should always use proper escaping, when building the queries.

This is how your code can look after fixing:

global $wpdb;
$value = $wpdb->get_var( $wpdb->prepare(
    " SELECT column_name FROM {$wpdb->prefix}plugin_table WHERE ID = %d ",
    get_current_user_id()
) );

The things you have to take care of:

1. You have to change column_name to proper column name you want to get.
2. You have to change plugin_table to proper table name you want to get the value from.
3. You have to change ID to proper name of column that stores the user id in given table.