Heavy Weight vs. Low Weight/High Rep
Both heavy weights and low weights with high rep lifting will make you stronger, the difference is in which type of muscle fiber is being recruited for the lift. The type of lifting dictates what type of fiber will be trained, thus determining what type of strength you will make gains in. Heavy lifting primarily trains your fast twitch muscle fibers, whereas high repetition lifts work more of your slow twitch muscle fibers. A fast twitch fiber cannot be trained to become a slow twitch fiber and vice versa, so the specificity of your training has a direct correlation on the kind of strength you are building.
If you are looking to improve explosiveness, overall strength, and bone strength, heavy lifting is what you should focus on. As an added bonus, heavy lifting actually improves your liver health, and has the highest calorie output ratio! Common sports that require this type of training include football (American), field events (shot put, broad jump, high jump, etc.), and shorter duration races (100m, 200m, 400m), among others. The repetition range for heavy lifting is 1-5 reps. I suggest staying on the higher end of the range (3-5) to avoid injury.
LOW WEIGHT/HIGH REP LIFTING
Contrary to popular belief, this type of training does actually help improve overall strength. The difference is that heavy lifting does so at a higher rate. There are, however, several benefits of high repetition training. The main advantage of this training is the improvement of muscular endurance. High rep training is also great for toning, which is important because lean body mass burns more calories throughout the day than fat mass does. Athletes that follow this protocol include long distance runners (5K races, marathoners, adventure races, etc.), soccer players, swimmers, and bicyclists.
Most people will need a mixture of heavy lifting and higher repetition lighter weight lifting in their training protocols; however, athletes training for specific endeavors, or people looking to improve a significant weakness or deficit in either system are an exception. As always we suggest you consult with a well-qualified trainer to meet your goals and coordinate a proper program. Especially where heavy training is involved an extra set of eyes to keep proper form and a spotter are essential to your safety and success.