Is Apple's Airport Express my solution?

I need to have a better signal and range in the second floor since my router is in the basement. I mainly need it for my wireless devices iPhones, iPads, iPods. I was watching youtube videos on apple's airport express but mostly of them are connected to a time machine? I don't own that, is that necessary? I'm not quite sure how the airpot express works, can it work with a router? Is it even going to fix the problem i have with my wireless devices going slow because of the distance of the router and the second floor?
does it work with non apple devices like PCs, tablets, androids?

can anyone give me as much info as they can please!

Added (1). So i can't use my verizon router?

Added (2). @ryan ^

If you are looking for higher wireless speeds, I would recommend using a hardwire Ethernet connection and then extending the network that way. Wireless to wireless extension can lower streaming and download rates just as much as poor signal can. In my house, I use an Airport extreme from my modem on the second floor ad then an Airport Express to extend my network in an area that I have had poor reception. I have tried both with and without a wired Ethernet connection and I have noticed significant changes to my download speed. I always have signal and of course the apple family of products has yet to fail me, but I have noticed this interesting difference. Yes all Apple routers and the airport express will work with non apple products. For the question about the time machine. You can use any apple router with the airport express (airport extreme and time capsule). The difference between the two is the airport express doesn't have any storage, so you can't store any files on it.

Edit:

If you want to extend your Wi-Fi coverage you want a network extender. You can use the airport express with any router, but the downside is the price is a little high. Google network extender. If you the airport extreme there are many advantages to it because you have apple products like the usb port for storage and the audio jack so you can connect speakers to it and turn those speakers into an airplay device and be able to play your music wirelessly to those speakers.
Here is a link to a network extender from
Bestbuy: http://www.bestbuy.com/...Id=2733324

Hope I didn't confuse you
Ryan