Food that is cooked by someone in front of you
Teppanyaki, literally means iron grill and can include okonomiyaki in its definition, but it generally refers to meat or seafood cooked over a grill in a high-end restaurant. In this type of restaurant, you can sit at the counter and see the chef carefully cook all the ingredients before you.
Robatayaki is fish or vegetables cooked in a fire hearth at the center of the restaurant. You can also sit at a counter and see the chef cook the ingredients over a charcoal fire, which gives them a delicate BBQ taste.
Kabayaki is eel dipped in soy-sauce and held together by skewers, which is cooked over a grill. Eel is often during the summer in Japan because it is said to help summer fatigue.
Yakitori is different chicken parts on a skewer over a charcoal fire. At casual restos, people sit at a table with their friends, but at tiny restaurants, people sit at the counter and can see the chef grill the skewers. In recent years, some more high-end yakitori restaurants have popped up, where you can have yakitori in a more Western setting and served in wine.
Food that you cook at your table
Shabushabu/ Sukiyaki (しゃぶしゃぶ/すき焼き)
Shabu shabu and sukiyaki are both meals that you cook in a hot pot at the centre of your table. Both are basically thin slices of pork or beef with vegetables that you cook yourself. The main difference is that for shabu shabu, you slowly insert the ingredients and cook them as you want to eat them and then you dip them in a ponzu or sesame seed sauce. Sukiyaki is generally already in your hot pot and is cooked with a sweet soy sauce.
Shabu shabu is normally cooked by yourself, whereas, sukiyaki, at some high-end restaurants would be cooked for you by the staff.
Okonomiyaki/ Monjayaki (お好み焼き/もんじゃ焼き)
When asked to describe these two dishes, Japanese will generally say okonomiyaki is like a Japanese pizza and okonomiyaki is a messier version of the latter. In my opinion, it isn't very similar to a pizza, it is more like a savoury pancake with tons of ingredients inside (seafood, mochi, pork, etc.) and topped with mayonnaise, Bulldog sauce and bonito flakes.
When ordering these dishes, you would get the ingredients already mixed in a bowl, which you would mix to a nice consistency and cook by yourself on an iron plate.
Yakiniku, also called Japanese BBQ, is pork or beef (and in some instances, chicken) which you cook by yourself at your table on a charcoal grill. It is up to you to make sure the meat doesn't overcook, but even if you have zero BBQ experience, it is quite easy but mostly incredibly tasty.