I'm making some assumptions, first that the reader is interested in improving his or her speed and performance. Second the reader is not an elite caliber endurance athlete, and finally that the reader probably has limited time available for training. My advice might be substantially for those who do not fit these assumptions.
What exactly is fitness?
I'm starting with the question of what is fitness, because there is a lot of confusion about what it means to be fit. It's not an easy question to answer, because there is no standard definition of fitness. Traditionally it has been defined in relation to an activity. Most people would consider a world class marathon runner to be fit, but is a world class marathon runner well suited to carrying a heavy man out of a burning building? By the same token, a world class Olympic lifter is also considered fit, but is an Olympic lifter well suited to competing in the Tour de France? So which one is fit? The answer is that they are both fit within their specific domains. I used these examples, because a marathon runner and an Olympic lifter fall on extreme ends of the spectrum of physical performance, so their fitness is very specialized. Could a marathon runner improve his or her performance by doing some Olympic lifting, or could and Olympic lifter improved by some endurance work? It's not absolutely clear, but there is some evidence to suggest that they might.
A case for general fitness and cross training
I'm of the opinion that the body is a system, and a weakness in one area can effect performance in a seeming unrelated activity. Although it seems popular now to criticize CrossFit and its training principles, but they do have a very elegant definition of fitness ( information about CrossFit's definition of fitness). The developer of CrossFit, Greg Glassman, has identified 10 elements of fitness. By consistently working to improve all 10 elements of fitness over a full range of functional movements, and over varying periods of time, CrossFitters have been able to produce a great deal of evidence suggesting that improvements in one area of fitness, often seem to produce improvements in seemingly unrelated areas of performance. Of course, this idea of "total" fitness and the importance of cross training are not new or unique to CrossFit. However, CrossFit has done a great deal to demonstrate that improving general fitness clearly helps to improve performance in specialized activities like running.
The bottom line
To reach your potential as a runner, add some cross training to your routine. By cross training, I specifically mean training in elements of fitness that seem to be unrelated to running. Train different movements, and train develop power, strength and flexibility. Also, I see too many distance runners who think that cross training means taking up long distance biking. While technically this is cross training, there is too much overlap between these activities to be effective in this context.
If you have read my other posts on training errors, you know that I generally don't recommend running every day. On the days that you don't run, either rest or do some type of cross training. For most people I recommend something along the lines of 2 to 4 days of hard running, with 2 to 4 days of cross training, and 1 or 2 days of rest per week. Finally, take your cross training as seriously as you take your running. Don't just do token workouts. Reaching your potential in anything requires more than just checking off items on list. You need to apply yourself to every aspect of your training, to reach your full potential as runner.
For further reading
What is Fitness? By Greg Glassman
My other posts on training errors