The FTSE 350 Index index is a market capitalisation weighted stock market index incorporating the largest 350 companies by capitalisation which have their primary listing on the London Stock Exchange. It is a combination of the FTSE 100 Index of the largest 100 companies and the FTSE 250 Index of the next largest 250. See the articles about those indices for lists of the constituents of the FTSE 350.
The FTSE 350 is not very widely followed. The terms "FTSE 350" or "FTSE 350 company" are however useful for referring to large listed British companies when one wishes to include companies outside the top 100.