The push press is one of the best exercises to develop strength and power because it allows you to use heavy loads but is also an explosive lift. It can also be used to gain muscle mass on the shoulders and upper back.
The unique benefit of the push press is that it allows you to use higher loads than you'd use in the regular shoulder press (barbell overhead press) and consequently you can use it strategically to overload your shoulder complex and upper back.
Leg and Hip Drive
With the push press, you use the biggest and strongest muscle groups in your body (the legs and hips) to set the weight in motion and use your deltoids and triceps to lock the weight out over your head.
However, in order to use higher loads, you need to be able to execute the exercise correctly and make efficient use of your hips and legs to drive the weight up. Therefore you need to have the bar in the front rack position supported on your shoulders.
Too often, you see athletes holding the bar with their hands, similar to the barbell shoulder press or military press. This position won't allow you to effectively transfer the forces from the leg drive and hip extension into the bar, and you'll lose some of the forces you've produced.
This means you won't be able to use the loads in the push press that you could potentially handle. And less weight used means less strength gains and less gains in muscle mass.
Are You Strong?
How do you know whether your push press strength is sufficient? You can look at the ratio between your strict shoulder press and your push press.
What I've found is that this ratio should be somewhere between 125% to 140%. This means your push press weight should be 125% to 140% of your shoulder press or military press weight, which aligns pretty well with the standards outlined in Know Your Ratios, Destroy Weaknesses.
In order to reap the benefit of the push press, you need to be able to execute it correctly, and that means to use the front rack position to effectively transfer the forces from the leg drive and hip extension to the bar.
If you're not able to use 125% of your shoulder press in the push press, you need to have a closer look at your technical execution.