With reading the link provided, I'll say this:
covered shares -- the broker is supposed to track cost basis and provide cost basis to the IRS on your 1099B when you sell shares. That doesn't mean the cost basis is correct, so one has to make sure the cost basis is correct. Essentially all shares purchased after certain dates should be covered and the cost basis will be reported to the IRS when shares are sold.
non-covered shares - shares purchased before a certain date do not need to have their cost basis provided by the broker to the IRS, but the cost basis will need to be provided by the seller to the IRS. However, many brokers have tracked the cost basis even before there was a rule for them to track it. The broker just does not have to provide the cost basis to the IRS. The seller needs to provide the cost basis when they fill out their Form 1040 Schedule D and the forms that lead into that. Once again, the investor needs to keep their own records since the broker may have incorrect records of the basis.
Basically, are the shares covered by the new rules or not?