Some form of Kiai is found in every martial tradition. There are differences in the Kiai, sometimes very subtle ones, different styles or types of martial art. There are great differences in the vocalizations and non-vocalizations found in the various martial arts and the Kiai used in Kendo has its own unique characteristics.
Explaining the various aspects of a proper Kiai, and how it affects a person's practice and why it is important is very difficult. A Kiai is a physical world manifestation of a phenomenon that takes place within the mind, heart and "Hara" of the Kenshi. It is not possible to separate the person from their Kiai; it is a reflection of who they are. A person does not learn to perform a Kiai, rather they learn to discover and manifest the Kiai that is personal to them and their spirit.
This is a confusing subject and this is only a limited explanation, so please understand there is more to learn beyond what you find here. Also please understand this is one of the most misunderstood subjects in all of the martial arts world. It is hard to discuss, particularly in English, without sounding like some sort of "new age guru".
As a beginner, starting the process of learning your Kiai, you need to learn to make three basic sounds from the Japanese language: "O", "Ri", and "A". Ask one of the Dojo Sensei how to properly pronounce these syllables. When combined, they form the basic sounds a beginner should make when making a Kiai.
OOOOOORRIIIIIIIAAAAA !!!!!! This is the basic Kiai of the Kendo beginner. These sounds have several practical functions. The pattern causes the user's mouth to open at the right time allowing for a high volume shout, and it allows for a long lasting Kiai. This, combined with where these sounds resonate from in the human body, requires the person to stand with correct posture and employ the correct stomach muscles in making the shout. This, in turn, trains the body to breath correctly during Kendo practice. On the most basic level, proper breathing allows for good oxygenation of the body and brain. This allows for not only good physical movement, but also good cognitive processes. If you are thinking clearly, you will make better choices during practice and be able to absorb more instruction, more thoroughly. As you repeat this type of Kiai, over and over, it will affect how you think during practice. Beyond the beginners stage, it will calm the mind, project the spirit, focus one's energy, and help bring the body, mind and spirit into harmony.
A Kiai made during the execution of a strike should be continued until the strike is completed. For a beginner, this is after he has passed the opponent and moved beyond his opponent's reach, beyond his opponents Maai. This may at first seem to be too long a distance, but it is not. When you breathing stops, your body will stop. Said differently, when you stop the exhaled shout of the Kiai, your body will stop, or, at the very least, lose the spontaneity of the movement. In addition, the pitch of the sound you make during a Kiai should rise and not fall. In other words, at the end of your Men strike, when you turn to face your opponent, the pitch of your voice should be higher Than when you started the Kiai. If the pitch goes down, so too will the movement lose its fluidity and smoothness. Ask one of the Sensei to demonstrate this concept for you.
Finally, as you progress in Kendo your Kiai will change. At first you should try to fill the whole room with sound, lots of volume and be loud. The more senior Kenshi and the Sensei of the Dojo will have lower, more focused Kiai, that will be sharper and more directed at the opponent. You will develop this over time but do not try to force it. As a beginner, Kiai loud, Kiai long and Kiai with spirit.