What is the relationship between the phrases translated (NRSV) "your desire shall be for your husband" (Gen. 3:16) spoken by God to Eve and "its desire is for you" (Gen. 4:7) spoken by God to Cain?
Both are תְּשׁוּקָה tshuqah: longing, desire.
This is a rare word. It occurs only in those two verses and in Song of Solomon, where it's also about love (where, interestingly, it's used of a man for a woman instead of the other way round).
From what I can tell, the etymology is unclear. Strong's links it to שׁוּק shuuq "overflowing", the connection being reaching out beyond one's bounds, in this case to take hold of something.
Many commentators have written about the Gen 3:16 / 4:7 pair; here are some public domain commentators on Gen 3:16 and some on Gen 4:7. A common theme is that the kind of desire meant in 3:16 is a submissive one given that it's paired with the man's ruling over the woman. Thus, when God says sin's desire is for Cain, and that Cain should rule over it — this too is the same word between the two verses, מָשַׁל mashal — the point is that Cain should take advantage of the fact that sin can only do what he allows it. It's a good thing that sin's desire is submissively for Cain, because this is what allows him to master it.
Interestingly, a popular traditional approach is to read the opposite into this: it's a possessive desire, not a submissive one. If you look at parallel translations of Gen 4:7, you'll see things like "eager to control you", "desires to have you", "wants to rule you", "desires to dominate you", and so on. This approach presumably reads Gen 3:16 in reverse as well: Eve is not submissive to Adam, but desiring dominance only to lose it. (In line with this reading, at least one translation interprets the Hebrew preposition differently: the desire is not for you, but against you. That doesn't seem impossible, especially given how few examples we have of this word.)
A third suggestion is that the desire Eve will have is not so much submissive as intimate, simply for marital or conjugal union. If so, then in 4:7, the metaphor is that sin wants to be that intimate, that close to you — which as far as Biblical metaphors go is no small thing, but is described as "becoming one flesh". Paul may have this sense in mind when he talks about "sin living in me" in Romans 7:17.
One other more neutral reading is that there isn't a particular connotation to this desire, but that the word is as multivalent as in English. Sometimes it means intimacy, sometimes submission, sometimes possession. This would be to weaken the semantic connection between the two verses — which is not impossible, since the Hebrew Bible does sometimes use wordplay for the sake of poetic echo rather than to say the same thing twice.
So there's a fairly rich history and multiple options to take there. I can't really bring you any farther without pretending to know more than what my cursory research has found.