Just a note: there are different ways to pronounce 'och'. It's usually pronounced with the 'ch' silent, as the robot voice does here, so that it sounds like the Swedish 'å'. You'd say it with the 'ch' like a hard k if you're speaking articulately/slowly, or when you want to put emphasis on the 'och': "Jag gillar katter OCH hundar!" ("I like cats AND dogs!")
There are a couple of different ways to pronounce "ch" in Swedish but on its own it is always a pure consonant sound.
They represent different grammatical genders. "en" is common gender; "ett" is neuter, both mean the same thing. It's like the distinction between masculine and feminine in Spanish.
I notice that the 'v' of 'brev' is slightly more distinct in its voicing: is it largely down to the fact that the following word begins with a vowel? Also 'ch' of 'och' is being voiced: for the same reason?
There's the word tidskrift which is more like 'journal', and a magazine can also be called ett magasin. But we're sloppy with these words, we tend to call all those things tidning in everyday speech. So all the versions are accepted – but if you want to be clear that you're speaking about en tidskrift or ett magasin, you should use those words instead.
If you are a native English speaker, then try to remember 'brev' by thinking of 'brief'. If somebody hands you a brief, it's going to a memo. This is much closer to a letter than a clunky newspaper.
How can I distinguish between en words and ett words? I mean, why is it ett brev and en tidning?
You can't. It has to be learned with the word, just like der/die/das in German or la/le in French.
Why is en and ett different for two objects? I thought en was for living...
Every word is either en or ett, and it has to be learned together with the word. There's no rule to it, but there are animate/inanimate tendencies in their usage. But it's still ett djur ("an animal"), but en stol ("a chair") for example.
I have a hard time understanding "tidning" when she speaks. Is she saying it the correct way?
Sounds okay to me. A lot of Swedes won't pronounce the "d" in "tidning" properly so it sounds like "tining".