that kind of behavior

"Behavior" is a person's way of acting. But the word "behavior" is mostly (but not always) used to talk about someone's negative actions:

“That kind of behavior would be totally unacceptable where I come from.”

What's going on with your behavior lately?

So the phrase "that kind of behavior" describes someone doing something wrong.

appropriate behavior

"Appropriate behavior" means doing what's socially correct. It means following rules of politeness.

Things that are not appropriate behavior in the U.S. include:

• Burping in public.
• Talking during a movie in a theater.
• Telling a woman that she's fat.

Some things that people do wrong are too serious to be called "not appropriate". For example, you don't describe beating someone up as "not appropriate".

(do something) in public

"Public" places are places where people who you don't know can see what you're doing. Some public places include:

• restaurants
• on the street
• at school

There are other places which might be considered public or private depending on the situation. For example, someone else's house is private for them, but might be public for you. On the other hand, if it's a close friend's house, you might not think of it as being public.

Describe things that you do in front of strangers as "in public":

I'm a decent singer, but I hate singing in public.