According to the BLS, the new index is referred to as the "Chained Consumer Price Index for All Urban Consumers," and is designated by the acronym C-CPI-U (as opposed to CPI-U for the regular CPI for all urban consumers). The BLS describes Chained CPI as follows:
The index employs a superlative Tornqvist formula and utilizes expenditure data in adjacent time periods in order to reflect the effect of any substitution that consumers make across item categories in response to changes in relative prices. The new measure is designed to be a closer approximation to a “cost-of- living” index than the existing BLS measures.
(Note that it's not actually important to know what a Tornqvist formula is in order to understand what follows.) So what does this all mean?