I am often asked this question and the answer is very much dependant on the type of exercises you do. A good weight training program can increase endurance, improve running technique, increase flexibility, decrease risk of injury, balance any muscular asymmetries throughout the body allowing your body to function more effectively, increase the strength of the body for going up and more importantly down hills, increase lactate tolerance,increase strength of tendons and ligaments, not to mention general health benefits such as increasing bone density, maintain muscle mass, increase metabolic rate, decrease blood pressure and having a positive effect on insulin resistance.
It is relatively easy to come up with a program to obtain the general health benefits but a lot more thought has to go into designing a program with the running benefits. It is not as simple as thinking that running works your thighs so any exercise that works you thighs must help.
There are a few basic principles to keep in mind. The first is the body's response to exercise is specific to the actual exercise performed. What this means is if you perform a squat for example then you will become better at squatting but just because it works the legs doesnt mean any of the benefit you gained squatting will transfer into running. If you want to get better at running then you need to pick exercises that challenge the body in a similar way to running.
A common exercise I am asked about is the leg extension ( sitting down, starting with bent legs and straightening your legs with a weight on your shins). If we look at this in a bit of detail I'm sure you'll understand what I mean. The leg extension works both legs at once, you are seated, your feet aren't touching the ground, your thigh muscles ( quadriceps) work to extend the leg. When you run, you land on 1 leg, and as your foot hits the ground it has to absorb all the weight of the body , your quadriceps work to slow the rate at which your knee bends. If they didn't work your knee would keep bending until you colapsed on the ground.
So one exercise starightens your leg in a seated position and the other slows down the rate at which your knee bends in a standing position. Those two desciptions sound very different.Consequently doing leg extension will have very little effect or carry over benefit to your running.
The second point to consider is that the body moves in three planes of movement. The three planes are forward/backward, sideways, and rotational. To understand this a little better think of a tennis player, when he runs forward he is moving in what we call the sagittal plane, when he goes sideways he is moving in the frontal plane and when he hits the ball his arm, raquet and upper body are moving in the transverse(rotational )plane.
On a smaller scale, every bone in your body moves in all three planes. Take running - as our foot lands our arch rolls in which causes our shin bone to rotate in, which causes our thigh bone to rotate in, our hip also drops on the opposite to our landing foot, this movements happen all the way up to our head. If we aren't able to control these forces then we lose power and efficiency and it leads to injuries. For example the inability of the lower leg and foot muscles to control the inward roll of the foot can cause shin splints.
When was the last time you ever did any lower leg rotational exercises to strengthen those mucles? When did you do any upper thigh rotational exercises to control the inwards rotation of the thigh bone?, what about any exercises to control the movement of the pelvis as it drops on one side when the other foot lands? Lack of strength in this area commonly leads to ITB problems.
The final point I want you to consider is the type of muscle contractions that running primarily uses. There are two main types of muscles contractions - concentric and eccentric. Concentric means the muscle is shortening as it is working eg as it does in when you straighten your leg in the Leg Extension exercise, eccentric means the muscle is lengthening whilst trying to work. This happens in almost every muscle in the body when you run. For example after your foot hits the ground your knee starts to travel forward and your calf muscles help slow this movement down so you can power off for the next stride.
Your exercises should emphasise the eccentric part of the movement as much as possible. You might be thinking well what happens when I push off onto the other leg, isn't that concentric? Yes it is but concentric action becomes a lot stronger when you begin with an eccentric action first. For example if you are standing up and I told you to jump as high as you can the first thing you would do is squat down (try it if you don't believe me). Why do you squat down when you goal is to jump up? Squatting down loads up the muscles eccentrically so now they have more strength concentrically. Think of it as stretching a rubber band. If you want to fire a rubber band across the room the more you stretch it first the further it will go. Our muscles work in the same way.
So when you are designing your next weight training program to help your running ask yourself - does this exercise look similar to running, does it challenge me in all thre planes of movement and am I loading the eccentric part of the movement first.