The shoulder joint is a complex construct which relies heavily on strong muscles and connective tissue around its joint to keep it healthy.
The overhead press, together with the bench press, is a major upper body strength movement - yet it is commonly not trained with the same frequency and determination as the bench press.
I believe one reason explaining this is that the overhead press is often wrongly attributed to be causing impingement syndrome - a painful shoulder condition which takes a fair amount of time to heal.
What happens during shoulder impingement?
The tendon of the supraspinatus muscle gets trapped in the space between the head of the humerus and acromio-clavicular joint (see foto below). This causes painful inflammation, inability to lift the arm beyond a certain point, and muscle spasms around the shoulder joint.
The major mechanism for developing shoulder impingement is repeated amounts of horizontal adduction and/or shoulder flexion combined with internal rotation of the humerus. Such movements include upright rows, front raises, as well the bench press (to a degree).
In the overhead press, a different thing happens: while shoulder flexion and internal rotation are happening to a degree, the shoulder SHRUG at the top the movement pulls the acromion away from the head of the humerus, creating breathing space for the supraspinatus tendon in that little gap. As such, the overhead press can NEVER cause impingement.
In fact, getting stronger on the overhead press develops a much healthier balance in the shoulder muscles compared to the bench press. In the latter, it is mostly the anterior part of the shoulder that is being strengthened: anterior deltoids, pectoralis major, and internal shoulder rotators.
In the overhead press, we are strengthening the external rotators eccentrically (by lifting the arms overhead). The anterior deltoids are loaded concentrically, while the lats and lower trapz are loaded eccentrically in the pressing motion. Furthermore, the triceps (which are important shoulder stabilizers) are getting a more thorough workout in the overhead press when compared to the bench press. Overall, this develops a healthier shoulder than the anterior-dominant bench press.
Does this mean we should never bench press? Of course not. It just means that the overhead press MUST be added to a bench press program to keep shoulder pain away.
The final key benefit of the overhead press is that all muscles being worked are strengthened in CONJUNCTION with eath other. They accelerate, stabilize, and decelarate movements TOGETHER. As such, doing banded external rotation movements in the shoulder (popular in rehab and "prehab" settings) does not strengthen the rotator cuff in its functionality, but only in isolation.
When you first start overhead pressing, choose a weight with which you can perform at least five reps with good form. As you put more and more weight onto the bar, your shoulders will get stronger by the day.
Stronger shoulders are healthier shoulders.