Penal Substitutionary Atonement is the view that Christ was a legal substitute for us on the cross and that he bore the penalty for our sins that is due to us. (Penal means legal as in penal-colony)  Jesus was made under the Law (Galatians 4:4) and bore our sins in his body on the cross (1 Pet. 2:24).  At his crucifixion, he was “pierced through for our transgressions” and received our chastening and scourging (Isa. 53:4-5).  What was due to us he bore by being made sin (2 Cor. 5:21).  This is exactly what the Penal Substitutionary Atonement is.  This is also called the Vicarious Atonement of Christ.

• Isaiah 53:4–5, “Surely our griefs He Himself bore, and our sorrows He carried; Yet we ourselves esteemed Him stricken, Smitten of God, and afflicted. 5 But He was pierced through for our transgressions, He was crushed for our iniquities; The chastening for our well-being fell upon Him, And by His scourging we are healed.”
• Mark 10:45, “For even the Son of Man did not come to be served, but to serve, and to give His life a ransom for many.”
• 2 Corinthians 5:21, “He made Him who knew no sin to be sin on our behalf, so that we might become the righteousness of God in Him.”
• 1 Peter 2:24, “and He Himself bore our sins in His body on the cross, so that we might die to sin and live to righteousness; for by His wounds you were healed.”

Penal means legal and, of course, “substitutionary” means that he took our place.  The legal aspect of Jesus’ atonement is evident in that He was made under the Law (Gal. 4:4) and he obeyed the Law perfectly (1 Pet. 2:22).  According to the Law, He was the sacrificial Lamb (Ex. 12:3; John 1:29), who as a Priest (Lev. 4:20; Heb. 7:23-24), who also made atonement (Lev. 17:11; Heb. 9:14).  He was without blemish (Deut. 17:1; Heb. 9:14), and He offered himself on Passover (Exodus 12:11-13; John 19:14-15).  So, he was a legal (penal) sacrifice.  His sacrifice was substitutionary in that the judgment that was due to us, he bore.  So, he was smitten of God and afflicted (Isaiah 54:4) as we should have been due to our sin.  He “was pierced through for our transgressions” and the “chastening for our well-being fell upon him” (Isaiah 54:5).  Notice that what was due to us, He took upon Himself.  Also, our iniquity fell upon Him (Isaiah 54:6; Heb. 9:28). Because sin is breaking the Law of God (1 John 3:4), and because Jesus equated sin with legal debt (Matt. 6:12; Luke 11:4), and because legal debts can be transferred, he then bore our sin when our sins were imputed to Him on the cross (1 Pet. 2:24), and He canceled the sin debt on the cross (Col. 2:14). Also, the death that we deserve (Rom. 6:23), he experienced (1 Pet. 3:18) on the cross.  It was there that He became sin on our behalf (2 Cor. 5:21) and offered Himself as a sacrifice to God the Father (Eph. 5:2; Heb. 9:14) so that we might be justified by faith (Rom. 3:26, 28).  His sacrifice was according to the Law (Penal), and He took our judgment (substitutionary) thereby satisfying the requirement of the Law of God and saving the lost through His legal, substitutionary atonement.

Though there are varying views of the atonement, the Vicarious Substitutionary Atonement (Penal Substitutionary Atonement) best explains the Scripture and most importantly, it probably relates the satisfaction of Law as a relates to the holiness of God.