A standing barbell shoulder press is slightly different from dumbbell shoulder press as it requires more muscles to stabilize the weight as your center of gravity is higher due to the barbell being press overhead. Furthermore, it requires you to balance the weight on both sides of the barbell which means you will be working your shoulders out more than you would with a pair of dumbbells.
Preferably, a standing barbell shoulder press should be done on a squat rack rather than carrying it up from the floor or a lower catch. The reason behind this is that there is a huge tendency for people to hurt their back when they jerk up the barbell, which could easily be avoided by using a squat rack. Similar to a barbell squat, make sure that the barbell rest on a catch which is slightly below your shoulder when you are standing shoulder-width apart. As for the handgrip position, it should be slightly larger than your shoulder width so that it can target the shoulders completely instead of the Trapezius when the grip is too close.
Let us now talk about how to execute the lift off the rack. First, tuck your elbows close to your body and point your elbows forward. Your hands should be in a position where they are above your shoulder. Squat down and absorb the pre-determined grip and stand up to lift the barbell off the catch. Take a few steps backwards and now you are ready to begin the exercise. Push the barbell slowly above with the barbell passing your face. After it has passed your head, press it upwards and slightly backwards such that you are standing directly under the barbell. Hold it with your arms fully extended for a second before you lower it down. Similarly, lower it down and slightly forward so that it will pass your face again.
There is a common practice of doing a standing shoulder press with the barbell behind your head. This is not a better technique to hit your deltoids but rather it is a dangerous exercise that could dislocate your shoulder joint from the scapula. Pressing the barbell upwards from behind also activates the Trapezius instead of the deltoids. The other thing to watch out for when you are pressing the barbell upwards is to note the position of your hips and your back. Both of them should not move forward or backwards respectively to engage the chest and the Trapezius to assist in pushing the weight up. If you found out that your form has degraded to this standard, please lower your weights so that you can reserve your back for more training to come in the future.