There is a lot to the subject of the atonement. However, one doesn't have to know all the deep doctrines surrounding this topic in order to begin applying the atonement to one's own life. One short, helpful article on Atonement is in the Bible Dictionary in the LDS 1979 edition of the scriptures. If you don't have those, you can go to the church's website here and click on The Scriptures. There, you have the standard works plus Guide to the Scriptures, Bible Dictionary, and Topical Guide. Back on the first screen of the church's website, you can click on Gospel Library and that will lead you to back issues of the Ensign and other church magazines, plus a host of church manuals. You can do searches, such as on Atonement, and find articles and lessons that will help increase your understanding. This is a very useful research tool for learning about any gospel topic.
Back to the atonement, the article in the Bible Dictionary is a brief and helpful description of the atonement and how it works. To apply it to your life, you must have faith in Jesus Christ, repent, and live the gospel fully. The latter includes the laws and ordinances of the gospel--the commandments in the scriptures. Part of this is to forgive others. When you recall the commandment to love your neighbor as yourself, it is implied that you forgive yourself as well. This is often difficult to do--we can be awfully hard on ourselves. Prayer and scripture study can help you, as can attending your Sunday meetings where you can learn much about the gospel.
What I want to say, though, is that it is not necessary to understand all the finer doctrinal points of atonement, or any other gospel principle, before you begin applying it to your own life. We grow in the gospel step by step, line upon line, precept upon precept. Don't wait for the blessings that will come to you as you live the gospel as best you can with the knowledge you have--now, today.