courtyard (v.69) — in the palace courtyard. The trials before Annas and Caiaphas were in the buildings that surrounded it.
gateway (v.71) — the passageway from the courtyard into the street
The palace or court of the high priest no doubt was constructed in the usual way. A gated porch admitted to a central courtyard on one side of which was the palace of Annas, and on the other side the palace of Caiaphas. Opposite the porch a raised columned pavement formed an audience chamber. Here Christ would stand as a prisoner before the high priest, and thus Peter was enabled to see Him and He to see Peter (Luke 22:1). — Williams, page 727.
your speech betrays you (v.73) — his Galilean accent
curse and swear (v.74) — perhaps profanity, or maybe calling down death upon himself if he were lying
Peter remembered (v.75) — The Lord turned (from where He was on the balcony) and looked at Peter (Luke 22:61).
The Lord’s look was probably not one of judgment but of sadness and love. When He warned Peter of his denials, He told him:
“Simon, Simon! Indeed, Satan has asked for you, that he may sift you as wheat. But I have prayed for you, that your faith should not fail; and when you have returned to Me, strengthen your brethren” (Luke 22:31-32).
Peter failed, but his faith did not. He sinned, but his salvation was never in question.