California bankruptcy exemptions allow you to keep certain property and assets in a Chapter 7 bankruptcy. For example, they protect some equity in your home and car and allow you to keep jewelry up to a certain dollar amount, household items, clothing, and most retirement funds. In Chapter 13 bankruptcy, exemptions play a role as well -- they help determine how much you'll have to repay creditors in your Chapter 13 plan (the more property that is exempt, the less you have to repay).
California has two separate lists of exemptions, referred to as Exemption System 1 and Exemption System 2. You can choose to use one or the other, but cannot mix and match between the lists. Which exemption system is best for you depends on the property you own and what you want to protect.