Lenses work by bending light. Light changes direction when it crosses the interface between two different substances, such as air/glass. This bending of light is caused by a change in the speed of light as it crosses into a different medium. This bending is called refraction. Refraction is quantified by Snell's Law. Refraction can be seen in real life by putting a pen into a glass of water. If you look closely, it will appear that the pen changes direction slightly when it enters the water.
Lenses come in two basic flavors: positive and negative. A positive lens will converge light toward a point on the far side of the lens and a negative lens will diverge light that same light. If the light is coming from a long way, ie. the rays from that object are parallel to each other or very nearly parallel, the light will focus at the focal point of a positive lens. This is the situation of burning ants with a magnifying glass. A 100mm lens will focus parallel rays of light 100mm from the center of the lens and a 25mm lens will focus those same rays 25mm from the center of the lens.
Since a negative lens diverges light, it will not form an image on the far side like a positive lens will. The diverging rays, if you track them backward they will converge on the front side of the lens, but will not form a real image. These lenses will have a negative focal length.
Focal length can also be measured in diopters (1000/focal length in mm). A 100mm lens can also be described as a +10 diopter lens. Shorter focal length lenses will have a higher diopter value and will bend light more than a longer focal length lens.
The focal length can be measured from either side of the lens - the lens has a front focal length and a back focal length. A 25mm lens is a 25mm lens whether I look at it from the front or the back. I can flip the magnifying glass over and still burn the same ant at the same distance.
The end result is that the focal length represents the distance that an object at infinity will be focused behind the lens. The focal length is a very important number and determines many aspects of macro photography such as extension, magnification, working distance and will regularly show up in the coming installments.