"Loro" means "they" by itself, and it makes sense when preceding a verb (they eat, loro mangiano). But "il loro" when followed by a noun (or noun phrase) means "their" as in the example here: "il loro cane". It seems strange at first to English speakers because we have two different words for them, but with a little practice it'll become second nature to you.
This page has more explanation about using possessive adjectives, along with many examples/translations and a table of all the forms (way down at the bottom of the page). http://italian.about.com/od/grammar/a/italian-possessive-adjectives.htm
In English we have possessive adjectives like your/her/his/their/my/etc. In Italian you need to add an article like il/la/etc. before them (except for family members*) and they need to agree with the gender and quantity (masculine/feminine, singular/plural) of the noun.
il suo cane = his/her dog
il cane = the dog
*Mostly, for a family member there is no il.
suo padre = his/her father.
However, for papà, babbo and mamma (affectionate names) you need the il/la. eg il suo papà, la sua mamma.
Also loro/Loro always needs the article and it doesn't change to lora for feminine words. Eg :
il loro padre = their father
il Loro madre = your mother
There are other rules. I've tried to keep it simple but at first it will probably seem confusing, however you will get used to it even if it doesn't happen in your native language. (What is your native language?)
Here is a link where everything is laid out in a table so it is easy to see how it works : http://www.unc.edu/~achamble/possessiveadjectives.html
I wouldn't try to remember all the rules at once if I were you, I would just learn as I go along.
In English we only say the rice if we are referring to some particular rice, if we are talking about what people eat in general we drop the article.
The woman tries the rice - means the woman tries a particular plate of rice. However to say "the rice" in English, is only possible if the person you are talking to knows what rice it is.
The woman tries rice - means the woman tries any old rice. Possibly this is the first time she has eaten rice in her whole life, possibly she has had trouble eating recently, so she is trying rice to see if she can eat it without being sick.
I don't see why you can't translate "Il loro cane mangia il riso." as either "their dog eats the rice" or "their dog eats rice" because there is no context.
I understand his/her confusion. In this particular example "Il loro cane mangia IL riso", the correct translation is "Their dog eats rice" - there's no "the" before "rice" even though in Italian sentence we have "il".
Would you say that's one of the cases where Duolingo is wrong or is there an actual explanation to that?
Thanks for all the help, btw!
Thanks for explanation! I could use both in this case but sometimes I can't (don't remember any example now) and just wanted to know why and what is the difference. But if this "rice" is any kind of rice - should we use "a" instead of "the"? And actually I thought that la/le/il etc. are equal to english "the" and the same is for "un" and "a". But now it seems more complicated to me.
un is a la/le/il you are right.
You can't say she tries a rice because rice is uncountable. You would have to say she tries a grain/type of rice.
That would not be general, that would be specific to that grain/type of rice.
If we take an example that doesn't have an uncountable noun so things are simpler:
She tries the oranges (She tries some specific oranges, maybe she is trying to see if they are edible or if they are ripe yet)
She tries oranges (She tries to eat any oranges, oranges in general. Maybe she has never eaten oranges before)
She tries an orange (She tries one single orange. This can't be oranges in general because it is singular and refers to only one orange.)
For English you might want to check out these links. They have better explanations and a short exercise to complete to make sure you understand: https://learnenglish.britishcouncil.org/en/english-grammar/determiners-and-quantifiers/indefinite-article-and
As for translation from English to Italian on Duolingo, you have to know that Duolingo doesn't always get it right, so on the occasions that you sometimes can't translate the/il it might just be a problem with the software.