In the admin, go to Tools > Redirection.
The screen you see is all of the site's current redirections. You can edit or delete any of these by hovering over the one you want to modify (options Edit, Delete, and Disable appear).
To Add a Small Number of Redirections
To add a redirection, scroll to the bottom of this screen.
The Source URL is the old URL that you want to redirect. The Target URL is the current URL you want the page to push to. The group should be redirections. Do all of this with relative paths. Hit "Add Redirect" when finished.
You can test to see if this is working by putting http://www.mydomain.com/old-broken-url/ in the browser and seeing if it pushes to the new URL. (substitute domain and url for the correct ones).