Black-and-white photography is based on black, white and tones of gray. How an image changes from a world of color to black, white and tones of gray depends on two things: how a specific color is translated into a tone of gray, and the inherent lighting contrasts of the scene in the first place.
I learned photography shooting black-and-white. Ansel Adams and W. Eugene Smith were my heros. Today, I've fallen in love with black-and-white again, largely because I began using Nik Software Silver Efex Pro 2. Download a full-featured trial version for free to see what it can do for you (www.niksoftware.com). Personally, I don't think any other black-and-white conversion software comes close.
Any good black-and-white conversion software—though perhaps a better term would be translation software—gives you options in how colors change to gray tones. Should a red be black, dark gray or medium gray? Should nearby greens be light gray, medium gray or dark gray? How about the sky? Should it be a dramatic dark tone that makes clouds jump out of it, or should it be more subtle to balance other tones in the photo?
These aren't decisions that can be made by a one-button fix. This is one reason why black-and-white can be challenging. Also, not all color images look good in black-and-white, no matter what you do. That also makes black-and-white work a true craft.
I'll often try out images quickly in Lightroom by going to the B&W tab of Develop, just to see roughly what they might look like in black-and-white. Then, if I like the look, I'll make my conversion with Nik Silver Efex Pro 2.
One thing I'm looking for—both while shooting and when reviewing images for conversion—is a basic structure to the photo that works in black-and-white. This is always related to light and dark contrasts. The structure of a photo is part of its composition, and is probably best described as being like the structure of a house or any other architecture. There are always some key parts to that house that keep it standing and give it its basic shape and form. The same thing happens with black-and-white, and I'll look for key parts of the composition that give the image structure and form in black-and-white.
Digital photography gives you a great way of doing this. Shoot RAW+JPEG. Shoot the JPEG as black-and-white so you can review it on your LCD to better help you see what's happening to a scene in black-and-white. Then when you open the RAW file, it will be in full color, and you can translate it into black-and-white with much more control than you could ever do in-camera.
Take lots of pictures and try to find images that work on your LCD as miniature black-and-white photos. That will help you start to look for and find the structure of a black-and-white image.