It is kind of a simple grammar actually... you use どんな in order to say: How?, What kind of? and, generally speaking, something related to type. On the other hand 何{なん} is used in order to specify characteristics of the modified term. However it gets a little bit more complicated sometimes, because it might be unclear where the difference is depending on the context. So, here we go with examples.
Describing content
As someone else pointed out, 何 is used to describe content. Consider the following:
それは何の本ですか。 => That book, what is it about?
この雑誌はファッションの雑誌ですよ => That megazine is a fashion one.
When content is involved, it is really easy because we are talking about books, megazines. But this will work for videos as well:
見た映画は何の(映画)でしたか。 => What was it about, the movie you saw?
Everytime you need to say some like to be about of..., no problems, use 何.
Describing places and people
It happens quite often to ask something like:
What kind of place is that?
Or:
What kind of town is it?
So, we can have the following:
Q) 東京はどんな町ですか。 => What kind of town is Tokyo?
A) あの、東京はすごく大きな町ですよ。 => Well, Tokyo is a really big city.
The same goes for people, have you ever asked someone something like: "What kind of guy is he?". Well, here we go:
Q) えと、前川さんの友達はどんな人ですか。 => Ehm, The friend of Maekawa-san, what kind of guy is he?
A) あのね、その人は優しいし、けいけんがあるし、いい人だと思う。 => Well, considering he is very kind and experience, I think is a good guy.
There is also the typical sentence about weather:
What's the weather like in Japan? => 日{に}本{ほん}はどんな天{てん}気{き}ですか。
Describing characteristics in general
何 is used also to specify elements that belong (physically or in an abstract way) to something. Consider the following:
その車は何色ですか。 => What color is that car?
何と言う意味ですか。 => What is the meaning of this? (quite an idiomatic sentence though)
何のためですか。 => What is this for? (asking about the purpose of a certain action)
Please note that here you want to have some sort of clear answer and a brief one as well. What color my car? Red! And that's it. I am pointing this because, in the next lines, I am going to give you a little hint about the good times when it is appropriate to use どんな.
In general
Not easy sometimes, but one general rule might be the following: Use どんな
whenever you want a description of something, when you want to know something more about a man, a place, a woman, an object. When you use どんな always expect your interlocutor to provide a brief/long description which will never be limited to few words. When using どんな you want to take some time and consider something more detailed.
Use 何for the rest.
Japanese language sometimes is not easy to grasp, and sometimes trying to be very firm and not flexible (trying to achieve a generic strict rule) is a wrong approach. I was among thos ones who tried that way, had to give up...