The focus of the story is Louise Banks (Amy Adams), a top linguist who is tasked with communicating with these weird aliens. And they *are* weird; both physically - some of the weirdest aliens I've seen in a movie - and mentally. The military wants to know, above all, Why They Are Here?
On the physical sciences side, we have Ian Donnelly (Jeremy Renner), a physicist who has the occasional useful insight. And the primary representative of the military is Colonel Weber (Forest Whitaker), who recruits them both.
The frame of the movie runs, roughly, from the aliens' arrival to their departure. But there's a prologue in which we see the birth, growth, and death in hospital of Louise's daughter, Hannah, and here the spoilers begin.
Through most of the movie, Louise has flashes of Hannah's life. These appear to be a flashback, but they are eventually revealed to be a flash_forward_; Hannah is a direct result of the alien encounter, and specifically of the Donnelly encounter.
S (my Beloved Spousal Overunit) theorized after the film that the whole plot of the aliens is actually a dream/hallucination/phantasy brought on by the death of Hannah. While I very much doubt that this was the screenwriters' intent, there are little details that are strange in context and easily explained by that theory; the primary "other" explanation is emotional flatness on the part of the characters.
Anyway, as Louise learns to read the alien language - speaking it is impossible - she learns to think like them, that is, in a nonlinear manner, and her brain gets (as Vonnegut put it) unstuck in time, she starts seeing the future - thus the flashforwards.
In the straightforward version of the movie where the alien encounter really happened, this makes Louise's decision to go ahead and have Hannah, knowing what will happen, profound. (Or is it a decision? Once you know the future, _can_ you change it? That gets into a version of the Grandfather Paradox...) In S's version, it makes the alien encounter a way of accepting the death of Hannah.
At any rate, I recommend _Arrival_ highly, and suggest you see it on the big screen while you still can.