"a tall neighbor"
Not sure if it is taught at this level since I'm not doing the course myself. However, this letter can be addressed as "velarized T".
Put your tongue in the position for saying "T" - the sound of "T" is typically slender, meaning the chamber of the mouth is narrow.
Now, pull the tip of your tongue just a bit back and press it hard against the roof behind the upper teeth ridge (this will cause the middle of your tongue to curve downward and thus making the chamber of the mouth bigger). As you do this move, eject air normally as you do when you say "T". Hope this description would do. If not, maybe some youtube videos would be helpful.
Yes, many people actually said that they hear this letter as "P" - but for me it sounds like velar-T. I think this is a brain trick. Probably people who never heard this sound before, their brains are trying to attach the sound to something they know and close to it to fill the gap of knowledge or identification of such a sound, and thus it attaches the sound of "P" to it. Just my guess
These are used here (I guess) to mark down some sounds that are not available in English and other European languages. While (t) ت is the regular English (t) sounds, (T) ط is a velar (T) sound (a sound made with the chamber of the mouth bigger than usual, or as some call it "broad sound").
Same thing goes for (h) and (H). While (h) هـ is the regular (h) sound as in English, the (H) is used for ح which is a fricative (pharyngeal?) sound, comes out from larynx or the are above that (not good with anatomy here).
Check Youtube, there are a lot of videos about Arabic pronunciation. However, be careful of those videos where they tell you that it is OK to say the letter ج as "G" or like the French (J). This letter in standard Arabic is, and must be, like the English (J) - as in Jacket.
Dear TJ, you were very kind, and skilful, in explaining the new letter T, but the fact is that Duolingo had already explained the difference between h and H. And pretty well, I think. I am not diminishing your excellent contributions, but pointing out that some people seem to have raced through the course without reading the explanations that precede each lesson. I am a plodder, and I read them. More than once.
I really don't know how the course was designed in Duolingo as I didn't go through it, but I wouldn't surprised if they provided a lousy explanations or even didn't give any hints at all. In the Turkish course you would go through 3 or 4 skills successively without any hints for each skill; They just throw words at you there; And I was thinking Russian was hard.
Anyway, I'm bearing in mind as well that the sounds in this course are made by a machine (speech machine or text-to-speech machine, or program) and on many occasions the sounds get corrupted or not correct. Some people here even reported hearing the (T) ط as (P), but probably this is a trick played the brain of the listener, but nonetheless, it helps to explain things as properly as possible.