r.e. Input goes to one of three: Accept(T), Loop(T), Reject(T)
The thing to remember there is Loop.
recursive is almost the same, but there's no loop.
So you need a language that loops forever on something for an r.e. (as well as being able to show the other two exhaust all the possibilities).
And with the recursive you must have one that can't loop eternally.
Is that enough to get you started?
Before you even think about that, I suppose the first thing to do is see if I'm talking nonsense again, eh? I seem to have gotten every question I look at back to front.
Just a thought. If you've done some question involving building a TM, you'll know how it behaves, right? So that would be one of your examples. If it has some eternal loop (like the perfectly good ba ba machine I was so puzzled about) then it's an example of an r.e. language.
Another thought. r.e. is the outermost circle of the Venn diagram, isn't it? (Better check that)
If it is, then you can take any language inside it, and you're r.e. Take some regular language, for instance.
Again, you might have examples lying around in the questions you've already answered.
In fact you probably just need an example of a recursive language? What do you think?