刚才 + [subject] + [verb phrase]
[subject] + 刚才 + [verb phrase]
刚才 + 的 + [noun]
刚才 means "just now, a moment ago", counted from the moment of speaking. It's a time word like 今天 or 昨天: it goes before or after the subject, and always before the verb.
刚 is an adverb. It goes after the subject only, and says something happened a short time before the moment you're talking about – which can be minutes, days or a year. 刚才 is always minutes ago, counted from now.
Because 刚才 is a time word, it does things 刚 can't. It goes before the subject, it takes 的 to describe a noun, and 不 or 没 can follow it.
刚 goes the other way: it can take a length of time after the verb, which 刚才 can't.
With 还, 刚才 describes a state that held a moment ago and has stopped.