If you want to put some related posts in a widget you can use the smart YARPP plugin. Or, if you merely want to list posts with matching tags, you can put a function in your Wordpress theme, or make your own plugin.
In some ways this is better than the fuzzy logic of YARPP because it gives you more control: you just give posts the same tag, and you know they are related. Plus, its easy to edit such simple code to suit (if you know how the loop works).
Try this function to start:
Also, for this to work you need to be able to put shortcodes in widgets, which does not work with the default WP installation. To enable shortcodes in widgets add this to your functions.php: