This piece, like My last duchess, is about a person who kills (or is about to kill) her rival, in the presence of her lover - who appears to be connected to the speaker in some way - perhaps her husband or an ex-lover who has spurned her for the rival who is soon to die.
It is in the form of a monologue, and once more the silent listener is important, too. He is an expert in poisons (like the apothecary in Romeo and Juliet) who sells his services to a wealthy woman.
The subtitle (ANCIEN RÉGIME) refers to an older form of rule or government - suggesting that the speaker comes from a past age.
We do not know for certain that the speaker is female - but this is suggested by the things, listed in the fifth stanza, in which she will carry her poison ("...an earring, a casket/A signet, a fan-mount, a filigree basket..."), and by her offering a kiss to the poisoner, when he has finished his…