What you said is about right, but it's supposed to be 'Ce sont les chiens dont j'ai peur' (no second 'ce', because that would be the equivalent of: "Those are the dogs that of which I am afraid'.)
That translates to "those are the dogs of which I am afraid" or "those are the dogs which I fear".
'Dont' translates to 'of which/ of what', and 'ce dont' translates to 'That of which/ That of what'.
A sectence with 'ce dont' : C'est ce dont j'ai peur' (That is what I am afraid of)
'Ce que' literally means 'That what', which is shortened in English to 'what'
I hope that helped :)
Here's a better response. http://french.about.com/od/grammar/a/indefiniterelativepronouns.htm
The idea is similar but saying "the one" makes it more specific. "C'est ce dont il parle" is literally (word-by-word), "It is this/that of which he speaks." Since that is a bit awkward in English, we would say, "This/That is what he is talking about". Your sentence would be, "C'est celui dont il parle."