One of the lessons you've just learned the hard way is that with any brand new memory card, it should be formatted in-camera before its first use. For what it's worth, I reformat my memory cards in-camera after each use and all the pictures are safely on my computer in at least 2 places.
As WillyG noted above, if you have the USB cable to connect your camera to the computer, use that method to transfer the pictures to your computer. If needed, you could also borrow one from any Canon user, as I think they all use the same mini-USB connector on the camera.
It's also possible that either the memory card is defective or the card reader you are using with your laptop. When I first got my 60D and discovered to my surprise that it used SD cards rather than CF cards like the 30D I had, I rushed out to buy an SD-to-USB card reader. When I couldn't read the card on any of my 3 computers, I figured it had to be the either the card or the USB reader. Using the cable got me by that time. I then bought a new card reader that was the same brand as my memory cards (Sandisk) and the same SD card read perfectly when plugged into that card reader.
So, it's possible that either the memory card reader itself is defective or the memory card. Trying your card in another computer with a different card reader (or one that's built in) might be successful. If so, then it's the card reader. If not, then it's the card.
So, for now, use the USB cable if you have it. Otherwise try a friends computer. Either way, be sure to format the card in-camera once all the pictures are safely on your computer.