I am also not learned, so please be patient with me too.
(1)..a basic concept.. There’s no “concept” here. Torah says in one instance of adultery to stone the transgressors. A law, din, like any other 613 one of them,
(2)..Is it Halachik.. Yes, when two witnesses come before sanhedrin of 23 and testify, and a host of other criteria are met, then beis din must act. In the case where stoning is required, they are directed to execute it. Not all cases of adultery warrant stoning, though.
(3)..why is this not inplace today?..Because we do not have sanhedrin today.
(4) ..at what point was it halted?.. I think at the destruction of the Bais HaMikdash (actually I think a bit before).