OK, after two days of beating my head against the wall I've figured it out: all I really had to do was to simply follow the Quick Start Guide (QSG) instructions, and my laptop has nothing to do with any of it. I'm going to explain what caused my confusion with this whole process.
I did not want to mess up and sacrifice an SD card or the RaspberryPI board just to see what happens, so lack of confidence was one issue.
Another issue is that there are more than one guide on the Web on what to do to get Raspbian going on the board (see the list below), and I did not know which guide to follow. (And I assumed that every guide would be different; I should have spent some time reading every guide and making sure they were suggesting the same sequence of steps. That's my fault.)
1)
http://www.raspberrypi.org/quick-start-guide ... or ...
2)
http://elinux.org/RPi_Hardware_Basic_Setup ... or ...
3)
http://www.raspberrypi.org/forum/viewto ... =91&t=4751
Also, because I was examining the Raspberry-branded SD card and noticed that it was acting like some ghost card that no OS on any computer can see, I did not know what to do with it. It already could have had the Raspbian OS installed on it. (I believe it was actually the board that had the NOOBS already stored on it that could be used to install the OS on the card.)
I did not know whether or not to use the SD formatting tool (as suggested in the guide if you are on Windows OS) to reformat the card. Strangely, after I plugged the card into my laptop, the Windows OS did not see the card (as would be expected) but the formatting tool did. It was at that moment that I've thought that maybe the card wasn't useless after all and I could just plug it into the board and proceed with further QSG instructions instead of ever having to plug it into my laptop (which I originally thought was a good idea if I wanted to store NOOBS on it).
Anyways, thanks to all of you for all your help. I really appreciate it.
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Quick tips for those who want to easily get Raspberry Pi up and running:
1) Do not use the HDMI-to-HDMI cable that comes with Raspberry Pi (I don't know why it did, and that was one part of my overall confusion). I once connected the board to my laptop, but the GUI output did not change; the laptop was still showing Linux (which I was using on my laptop at the time). Your laptop will not serve as a substitute. Use an actual monitor. So, you actually have to use an HDMI-to-DVI lead. I went to Best Buy and bought a DVI-D-to-HDMI cable, and it worked between the monitor and the board just fine. I bought the RCA composite video connector just in case but haven't really used it.
2) You can utilize the SD formatting tool to make sure that your card is actually usable, so that you wouldn't have to waste $13 on a brand new 8-GB SD card.