Judo is a martial art that primarily involves throwing your opponent. Judo translated into english is "gentle way". The throw O Goshi is a foundational throw in Judo that involves putting your opponent on your hip and tossing them in front of you.
Every throw in Judo has three aspects. Kazushi, which is the act of off balancing your opponent, Tsukiri, which involves putting yourself in position to make the throw, and Kake, which is the execution of the throw.
Step 1: Kazushi (off Balance)
In Judo the grip you take on your opponent is crucial. The standard Judo grip (for the right handed) involves taking your right hand and grabbing your opponent's left side lapel, just below the shoulders. Your left hand should be on your opponents left arm, just below the elbow. Standard Judo stance is having your feet a bit wider than shoulder width, with your weight on the balls of your feet.
Now that you have a grip, it is time to use it. The Kazushi (off balance) for O Goshi involves both your hands, as well as your feet.
With your hands you will want to both pull and lift your opponent. To do this imagine bringing your hands to your shoulders. This does not need to be a large movement, just enough to bring your opponent's weight over their toes.
The feet are important during this step as well. As you are pulling with your hands your right foot should take a half step in front of your opponent's left foot. Remember to be bending your knees during this step as it will benefit you as you go to finish the throw.
Step 2: Tsukiri (setting In)
Now that you have sufficiently off balanced your opponent, it is now time to position yourself for the throw.
The first thing that you want to do is bend your knees. At every step in the throw you should be bending your knees more and more, as the goal is to have your hips below the hips of your opponent. If you are taller than your opponent you are going to have to get very low.
Next you are going to bring your arm around your opponents back. Like in the diagram above, you will need to get your arm all the way around as far as possible. If you leave your arm shallow on the back, you will have a difficult time lifting your opponent.
Now for your feet. Like in the diagram below, you are going to want to have your feet within the frame of your opponent. Having your feet too far apart will result in loss of control. Your toes should be pointing the same direction as your opponent's. Your opponent should now be up on your hip, and you are ready to finish the throw.
Step 3: Kake: Finishing the Throw
Now that you have yourself in position to throw, it is time to put your opponent on the mat.
To finish the throw you will need to turn your head. An easy way of doing this is to try to look back at your opponent's feet. As you look pull your hands in the same direction, this will turn your shoulders which will take your opponent to the ground. As your opponent hits the mat, make sure to not let go of their arm. By holding on to the arm you keep control of your opponent.