You would be wanting what is called a range extender. Although, another simple solution would be to either purchase a router with a larger initial range, or have an electrician wire your house for an ethernet port downstairs. If you can't get any signal at all, even with a range extender it will be a much weaker signal.
Having the electrician run an ethernet link from the upstairs router to the downstairs location would probably be the best option. Wired is always better than wi-fi for gaming. You could also run it through vents or just down the stairs yourself if you want a less permanent and cheaper solution.
I've used a 100ft ethernet cable to run from the basement all the way up to the upstairs bedroom one time when I was desperate for my Xbox Live. Now, though, in my apartment everything is wired up by the TV so it's all right there.