"I have one apple."
It can mean "one", but the word 'ein' has to be declined to 'einen' in accusative. It does not matter whether it is an article or a numeral.
Theoretically it is correct, but it is a colloquial way to say that which is not accepted by Duolingo.
Still don't know the difference between nominative and accusative... that's definitely something that should be emphasized early on with these lessons. Not sure where to learn more about this.
You can always try Wikipedia: http://en.wikipedia.org/wiki/Accusative The article might be understandable even for non-linguists. Briefly: Accusative is the case of direct object. When you use a verb that does something with some other thing, for example haben (to have), it takes this other thing (object) in accusative. In this sentence “I” is a subject (the doer of the action of having, WHO has?), whilst “an apple” is an object (the recipient of the action – WHAT is being had?). And an object of such verbs (they are called transitive) is in accusative.