Hi,
Couple of interesting things here. Perhaps GreatLakes may require a more solid definition of kata, and for good measure, my take on kiai as well.
Kata.
Well, kata is a Japanese term that literally means "form", or "shape". The method of training refered to as kata training is what NLP practitioners refer to as "modeling". In this method, the student repeats the same movements and actions over and over without changing them, in order to instill the basic strategies and concepts of a martial system. As a result, you could even say that BJJ uses a kata method, in that you constantly repeat the same action (a particular sweep, for example) over and over in order to be able to use it at any time. The term used would probably simply be "drill", though.
In old Japanese arts, known as koryu (old styles), kata training often refers to paired combat exercises, with a specific attack and response scenario. This is applied to unarmed and weapon schools alike, with different schools having longer or shorter kata depending on the school in question. This type of kata training exists even in more modern systems such as BJJ and Judo.
In more modern arts (such as karate, tae kwon do, various kung fu systems), the kata/form/poomse approach is far more often a long string of movements practiced solo. This is, as stated, not the only definition of "kata".
Kiai.
Kiai is often translated as "Spirit(ed) Shout", but the literal translation of the characters is "Energy" (ki) "Harmony" (ai). These are the same characters as in aiki(do or jutsu), just in the opposite order, so the idea is kind of the opposite from aikido. In aikido, as I understand it, the characters read "Harmony Energy", giving the inference of harmonizing with the energy of the situation, the attack, the opponent, or some other such concept. Kiai, on the other hand, reads "Energy Harmony", and that gives the idea of harmonizing your energy (or intention) together with your actions to create an effect on the outside environment, be it a person, situation, or something else entirely.
This kiai is more a concept of directed intention, rather than just a noise made for show. Of course, it is a very good way, as mentioned, to develop proper habits in breathing when striking as well, and to that end, there are a number of different physical (verbalised) kiai taught. In general terms, there are two main types of kiai, offensive (ending in a vowel - Yah, Doh, Ei etc), and defensive (ending in a constanent - Hut, Hup, Ei-Yush etc). The offensive kiai end in a vowel, as that allows the breath to leave the body, allowing the transferance of energy freely without the inhibition that comes form tensed muscles. The defensive kiai do the opposite, ending in a constanent specifically to hold the breath in and strengthen (re-inforce) the body to aid blocking and recieving manouvres. For the record, though, kiai can be used in a number of other ways, such as fakes, but I'm not adressing that here.
Specific traditional schools sometimes have associated kiai specific to them; for instance, Gyokko Ryu has Yah, and Doh (with the breath held in for defensive use), Takagi Yoshin Ryu has Ei and Ei-Yush, but it's not really focused on. We tend more to explore the concepts and allow students to express their spirit as they feel it... although this usually is an offer taken by the more experienced.
One last thing on kiai: the physical (verbal) aspect is just one expression. There are also silent kiai, and what are known as "shadow kiai". The important thing is your reason (intention) behind the kiai. But when verbalising, make sure the sound is generated from your diaphragm, not your throat. If you have a sore throat afterwards, look to how you're doing it.
Kime no kata.
This is to expand on Tez3's mention of it in Judo. Yes, it is found in Judo training, but that is not where it is actually from, merely where it is preserved. The old story goes that a formal self-defence system, relatively simple and easy to learn was wanted, so members of various organisations go together to create this system so it could be taught to the general public. It is very Jujutsu in it's approach, however it is not specific to any one group or ryu. It would most likely have been lost as it lost importance, if it were not for the efforts of the Judoka in preserving it within their curriculum, which I appreciate greatly (I love the Kime no Kata, and teach it every now and then to give an insight into what Goshinjutsu was and is now, and how the cultural differences change what is considered important).
So I'm sure you can see from my words here, I believe that kata and kiai have great benefits, even if you don't use these terms, you are probably using the concepts anyway.