Inflation is a loss of fiat currency’s purchasing power over time reflected in consumer market price changes.
A visual representation of the dollar value decrease since 1913 based on BLS’ CPI and historical prices database (Novermber, 2021). Source:

The US Inflation Indexes

The US inflation is calculated and reported by the BLS (Bureau of Labor Statistics) and used by the Federal Reserve to guide their monetary policies.
The Federal Reserve (the Fed) is the US central bank, whose board members are elected by the president and approved by the senate. BLS is a statistics unit of the Department of Labor that supports the Federal Reserve with data and depends on Congress for its budget and directives.
The most important inflation metric and what you get when you search for the current inflation level in the US is the consumer price index (CPI). The CPI measures the price change of a representative basket of goods and consumable services over time. The current 6.5% inflation (as of Dec 2022) is the year-on-year (YoY) change in CPI, which expresses a difference in price increases between Dec 2021 and Dec 2022.
There are multiple types of CPI index and other supporting indexes that the Federal Reserve reports to help investors gauge inflation, i.e., core CPI (CPI without food and energy), chained CPI (CPI corrected for predicted and extrapolated consumer buying trends), PPI (producer price index), CRB (Commodities Research Bureau) index, money supply (M1, M2, M3, M4), the Fed's balance sheet, Treasury bond yields, etc.
The BLS CPI is also used across the US to negotiate wages and calculate contracts, GDP, retirement income, social payouts, stock dividends, etc.

The UK Inflation Indexes

The UK inflation is calculated by the ONS, an independent data aggregator with strong links to the government. The UK uses multiple CPI metrics that differ slightly from the US situation.
The UK CPI doesn't include homeownership costs but is more often compared to the US and the EU indexes. Within the EU, the countries often use 2 different CPI indexes. One harmonized index that uses similar methodology across all the EU members and allows for comparison and calculation of the EU index, and one CPI that is used internally. The UK CPI was previously used within the EU block.
However, the UK's CPIH which includes homeownership costs similar to the US OER (owners equivalent rent) is more similar to the US CPI and even more to the Truflation CPI, because the UK Truflation index includes more of the homeownership costs like interest rates on mortgages and other costs.