Welcome back, Apple! I have long regretted the circumstances under which you left the forum, as MT, in 2015. Please feel free to go back to being "MT"; I can see that "Apple" and "MT" are the same person. This will be good for the sake of continuity; you may want to research some of your old threads as MT. Although this account has threads, the last one was 15 years ago!
We can say, "Look at the word" but not "Look the word at".
However, we can say "Look up the word " and "Look the word up" or "Look it up".
Why is this?
I've read somewhere that "at" is a preposition but "up" is not but adverb, but this doesn't solve my problem
The term that is generally used for "up" in "look up" is particle. We also use "particle" for "down" in "hunt (something) down," "turn (something) down," etc.; for "over" in "think (something) over," "do (something) over," etc.; for "in" in "turn (something) in," "log (someone) in," etc.; for "out" in "try (something) out," "draw (someone) out," etc.; for "off" in "turn (something/someone) off," "hand (something) off," etc.; and so forth.
I almost always decide whether or not I can put the word between the verb and the preposition just by a hunch.
Could anyone tell me how to explain this point of grammar?
The best way to learn phrasal verbs (verbs consisting of a main verb plus a particle) is simply by reading and listening to English as much as possible -- you will, as you know, simply see and hear people separating verb and particle -- but you can also consult lists of phrasal verbs. You could even invest in a phrasal verbs dictionary, if you haven't already done so.
"Look at" in "look at (someone/something)" is called, not a phrasal verb but, rather, a prepositional verb. "At" is a preposition in that phrase. I actually can't think of any phrasal verbs with "at." Perhaps it cannot moonlight as a particle in addition to its prepositional function, like "up," "down," "off," "on," "in," "out," and "over" (all of which can be prepositions or particles, depending on the case).
You have already mentioned that we can't say "look it at," though we can say "look it up." This difference holds throughout, including with the above words that can be either prepositions or particles. For example, in "turn something down" (e.g. "turn an offer down" / "turn down an offer"), we can have that order; however, we cannot have that order in "turn down a driveway," in which "down" is a preposition, not a particle. We can't say "turn a driveway down."
Another syntactic difference between phrasal verbs and prepositional verbs is that phrasal verbs must have the particle after the object when the object is a personal pronoun. Thus, although you can say either "look up the word" or "look the word up," if we change "the word" to "it," you will have to say, "look it up." You cannot say "look up it."
With prepositional verbs, it's different. You can say "look at it." Similarly, when "down" is a preposition after "turn," as in "turn down the driveway," we can replace object of the preposition with "it" and say "turn down it." But when "down" is a particle after "turn," as in "turn down the offer," you cannot say "turn down it." Indeed, that would change the meaning entirely.
Apart from memorizing phrasal verbs lists, and apart from from reading and listening widely to English and becoming familiar in that way with phrasal verbs, you can simply think about whether the word following the verb has a literal meaning or not. In "look up a word," are you looking in a particular direction? No. If someone asked you, "Where you you looking?," you could not answer, "Up the word." But you could use such an answer if "up" were functioning as a preposition after "look," as in "look up the hill." "Where are you looking?" Answer: "Up the hill."
Before I go any further, I'd like to see whether you feel that you have a better understanding of it now. This is a BIG topic. People have written books about this, and I have at least two books on it in my grammar library. We could get into phonological stress-pattern distinctions, and strange passivization patterns, pied piping (you can't say "Up which word did you look?"), and other fancy things like that. But I have already spent far too much time composing this post. I mainly just wanted to welcome you back to the forum.