The answer here varies from runner to runner depending on the runner's long-term goals, and how much running is done during the week. For instance, a runner who runs 5-6 days may choose to focus only on strength training with minimal cardio/explosive moves to rest. But, for runners running 3-4 days a week, strength training could be combined with a non-running cardio activity such as rowing, swimming, or even hiking.
For all runners, core work, and work on the muscles in the running chain (glutes-hamstrings-quads) are beneficial no matter how long or how far you go. Check out this workout from Outside Magazine as an example.
For me, since I don't have access to a gym at the moment, I do simple lunges and squats to strength train on my off days. I enjoy pendulum lunges to promote strength and balance, and squats because I can do them right in my home. I don't have free weights, so sometimes if I want an extra challenge with squats, I'll fill two water bottles and hold them in front of me during the exercise.
I know that there are many runners in our community who have other ideas on great exercises, but hopefully this gets you started!