The standard way to measure inflation has been with the consumer price index (CPI). The CPI has been used to calculate annual adjustments to social security benefits, federal pensions, military and veterans' benefits, and tax brackets, exemptions, deductions, and credits. According to some experts, the consumer price index currently used overstates increases in the cost of living.
So how is the "chained CPI" different? It makes different assumptions about how people spend. An oversimplified example: If a severe freeze drives up the cost of oranges and orange juice by 20%, people are likely to switch to a cheaper alternative, say, apples and apple juice instead of continuing to pay the higher price for oranges. This keeps spending more level than the regular consumer price index would indicate.
A switch to the chained CPI would mean that those government payments linked to inflation would rise more slowly. Applied to the tax code, the chained CPI would mean smaller inflation adjustments to tax brackets and other tax numbers, resulting in higher taxes than by using the regular CPI.