What is a word for talk or conversation done by a child? The aspects I wish to portray is the way in which they speak continuously, in an animated and enthusiastic way, and often nonsensically. Preferably the word should convey this in a positive light.
Based on your comment above, I'd say the word you are looking for is babble.
- to utter sounds or words imperfectly, indistinctly, or without meaning.
- to talk idly, irrationally, excessively, or foolishly; chatter or prattle.
- to make a continuous, murmuring sound.
Crib talk describes a child's conversation with itself, often a bedtime monologue. It is a neutral, non-judgmental term.
@Jim's babble is a hot contender for the most common word we use to describe a child's speech. But OP specifically asks for a "positive" word, and for my part I have to say I normally see that one as somewhat negative - of the three definitions he gives, the first two are effectively critical, and the last seems primarily prompted by the fact that we always speak of babbling brooks.
I therefore suggest chatter. Skimming through a few instances of children's/childish chatter, it seems to me that although childish chatter is frequently used in negative contexts, children's chatter is normally used positively. The words children's and childish have opposing connotations, but chatter itself seems to be either neutral or positive in most contexts.
A somewhat similar concept in psychiatry is confabulation sometimes also called "honest lying" in which someone is driven to carry on a seemingly normal and earnest conversation that is in truth totally disconnected from reality without an intent to deceive, simply because they are neurologically compelled to do so. See e.g., Sam Kean, "The Tale Of The Dueling Neurosurgeons" (2014) at pgs. 272-277, 323-324, 342 and 377. The term has a neutral to positive connotation that distinguishes it from compulsive lying with ill intent.