冷たい means cold to the touch, 寒い　means cold and 涼しい is cool.

Again, native speakers please correct me, but personally I feel は almost never corresponds to "a" in English, precisely because it tends to be used only after the thing has been brought up already. So there often is a "the" or something else like a possessive pronoun. For example:

• 私は犬と猫を飼っています。犬は白くて、猫は黒いです。'I have a dog and a cat. The dog is white and the cat is black.'

But if you ask me that's more a coincidence rather than は or が corresponding closely to English articles. It's a by-product of the fact that は tends to mark definite noun phrases and those are often marked with "the" in English (although it could also be something else, like a possessive or demonstrative pronoun). Also は can mark indefinite nouns, in which case we would usually be talking about a general category which the listener can be expected to know about, for example:

• 猫は鼠を食べる。 'Cats eat mice.'

が corresponds to the indefinite article even less because it doesn't have as much of a restriction in that regard than は. It simply marks a subject which is not the topic and can be definite or indefinite:

• 私は妹がいます。'I have a sister.' literally: 'As far as I am concerned, there is a little sister.' ("I" is the topic and "have a little sister" is information about that topic. So "little sister", being the subject of います 'exists' but not the topic, is marked with が.)
• 彼女は髪が長いです。 'She has long hair.' literally: 'As far as she is concerned, the hair is long.' ("She" is the topic, "hair" is just the subject of "(is) long".)*

So the main question you have to always ask yourself is: What is the topic, the main focus of what I am saying – the thing that I am providing information/asking about? That thing will be marked with は. Conversational flow usually dictates that that thing has to be something that the listener can be expected to already know about – a lot of the time either something which has come up before or a general category. If there is a subject in the sentence which is not the topic, it will get が.

I hope that helps.

* Of course, the last English sentence could also be translated literally as 彼女の髪は長いです。 'Her hair is long.' The difference is that in this case "her hair" as a whole is the topic, so I would expect that the next sentence will also be about her hair. In the previous version with "her" as a topic you could go on describing other things about her – or you could make "hair" the new topic in the next sentence if you want to.

Not native speaker, but been practicing with native Japanese speakers. Wa and ga don't translate to articles. These are like markers. Now that I think about it, I don't think there's 'a' or 'the' in Japanese. wa = marks the topic

What I do to make it easier is:

wa= as for this, about this Watashi wa John desu - As for me, it's John.

Watashi no namae wa John desu. About my name/As for my name, it's John.

ga = object of adjectives (there are probably other uses, but for beginners, it'll help us get by). - x GA y, I think of y as something​ that would describe x in some way

It also helps to pay attention to what words it usually comes with (ex. ga arimasu, abilities/can do like hanasemasu)

Inu ga ooki desu. The dog is big.

Inu ga imasu. The dog exists/is existing. = There is a dog.

*I think of this as if it's describing that the dog exists

Now, compare it with: Inu wa doko desu ka. About the dog, where is it? = Where is the dog?

I am not native but the way it has been explained to me is that は is used to introduce a new topic. Whereas が is used to identify the topic when it is unknown. i.e. If I asked "Who ate the chicken?" I would say "誰がチキンを食べた?" and the other person would respond with "アリスが食べた." They respond also using が because the current subject is not Alice it is the person who ate the chicken. If you were to say 誰はチキンを食べた. it would translate as "did who eat the chicken." Which makes little sense.

Maybe but the question is what would be the alternative? If you want to match English syntax onto the corresponding Japanese sentences, you have to conclude that が sometimes marks the subject and sometimes the object (私は犬が好きです "I like dogs"), while は usually marks the subject but sometimes the object (映画はあまり見ない "I don't often watch movies), the possessor (彼は髪が長い "His hair is long"), or something which doesn't have any more than a loose hazy connection to the clause at all (僕はだめです "As for me it won't do"). Is that really more helpful?

I think the better solution is not to try and fit the Japanese sentence into the English pigeon hole but rather do one’s best to understand how Japanese sentences work. So the above sentences become:

• 私は犬が好きです “as far as I am concerned, dogs are likable”
• 映画はあまりみない “as as movies are concerned, [I] don’t watch [them] very much”
• 彼は髪が長い “as far as he is concerned, the hair is long”
• 僕はだめです “as far as I am concerned, it won’t do

True, the Japanese structure does take a lot of getting used to (and the concept of “topic” in particular is indeed something fairly foreign to European learners; I took quite some time getting used to it as well. But once it clicks you realise that it’s a really neat thing to have!) And maybe it does help to take a middle approach and learn along the lines of “the Japanese equivalent of the English construction x is y, which literally means z”. But if you ask me, if you try and view Japanese exclusively through the lens of English, you’re doomed to failure.

Please note that が is used for non-topic subjects. The main problem, I think, is that Duolingo never gives us enough context to tell us what is the topic.

It is NOT the same. In your version　つめたい　みず　です - the adjective is directly modifying the noun - thus it means It is cold water. In the original sentence (where either は or が could be used after みず, but to be consistent we'll use が as in the original sentence) - が after みず tells us that みず is the focus of the sentence, and that that's where our translation should start. It is a simple A is B type sentence where が (or は as the case may be) tells us that this sentence is telling us some information about the みず - namely that the water is cold. The difference is that in your sentence it is already established that the water is cold - it is a statement of fact. Maybe someone wanted to make some tea and thinking the water in the kettle was hot poured some into a cup only to realise that it is cold water - not hot water and so they need to pour it back into the kettle and boil it. In the original sentence someone is telling someone else that the water is cold, possibly in response to someone inquiring after the water temperature - so they are giving information that is new or correcting someone or confirming that the water is cold.