To atone for something is to make amends for an offense. Often this carries a legal sense in which a payment is made for a crime. To redeem something has a similar meaning. For example, to pay a ticket for an impounded car would be to redeem the car. Paying the price would bring redemption.
Propitiation is a similar word, and suggests an appeasement of an angry party.
All of these terms are used to describe the death of Jesus as a payment for sin. Thus, God is angry with moral agents, such as humans, and He has allowed His son to be killed to atone for sin. Jesus has redeemed humans. His death acts as a propitiation for sin.
The entire Old Testament system of animal sacrifice, found in the law and in the Jewish temple religion is identified in the New Testament as a foreshadowing of the atoning work of Jesus.
Atonement allows for reconciliation between man and God. It allows a forgiveness of sin that is substitutionary. That is, Jesus substitutes for us and dies in our place, thus purchasing our freedom. God then becomes our friend and we have free access to His presence.
These great themes underlie the “good news” of Jesus. Saying that we are “saved by grace” often sums up the work of atonement and redemption.
It is worth pointing out, though, that while this work of Christ allows us to undergo conversion immediately, sanctification and growth in Christ are based on our relationship with Him and is not automatically produced as part of the atonement. One of the reasons why contemplative and monastic ideas and practices are growing in protestant circles is because of the failure of evangelicalism to address the need for believers to go beyond the open door of justification into obedience.