Dynamic range is simply the ratio between the highest and lowest light intensities a.k.a. black and white. A photo with a wide dynamic range has both extremely high and extremely low light intensities. A silhouetted subject is a perfect example of this. On the opposite end, a narrow dynamic range has light intensities that don't veer too far from one another. The exterior of a building on an overcast day is a perfect example of this.
A camera's sensor can only handle a specific dynamic range. Pushing beyond that range with a light intensity that is too bright, too dark, or both will cause areas of your sensor to loose light information. In other words, you will have non-recoverable parts of your photo because they're beyond what you camera's sensor can handle.
The human eye, on the other hand, has a greater dynamic range than any camera's sensor. That is why when you look in a room with a window in it on a bright day you can see both inside the room and out at the same time, but if you take a photograph of that room either the room is too dark or everything outside the window is blown out.
The key is to find a light intensity range that is within your sensor's dynamic range, but how do you do that? That is where the first feature most people hide when they get a new camera comes in handy... The histogram!
A histogram shows you the range of brightness levels in your shot from black to white. The shape of your histogram is entirely up to you. There are suggested histograms based on the scene your shooting, but that may not be the way you want to shoot. The main reason to look at your histogram is to avoid going past the far left or far right edges.
When you edit a photo, you can always adjust the shadows, highlights, and exposure as long as every part of your photo fell within your sensor's dynamic range. If you went beyond that range, your adjustments will be limited because the sensor couldn't capture any light information due to parts of your photo being too bright or too dark. Here's an example: If you take a picture at night with the moon in your shot, but your histogram shows the moon's light intensity passing the right edge of your histogram you wont be able to recover most details in the moon. However, if your histogram goes up really high on the right hand side, but doesn't pass the far right edge, you should be able to adjust your highlights until the moon shows it's face.
Hopefully now you understand what a dynamic range is and how to make sure you always shoot within it.
Best tip I can give you is to ALWAYS shoot in RAW. RAW images capture a wider dynamic range and will allow for much greater adjustments.