The API number is a number the American Petroleum Institute uses to identify oil and gas wells throughout the country.  The number is 14 digits which are essentially broken into four parts.
Example: 21137441900000

The first two digits are 21 which is the code for Michigan.

The next three digits are the API county number, starting with 001 for Alcona, and stepping up by two for each county until the last county, Wexford, number 165 is reached.  In the example the 137 represents Otsego county.  There is a table of counties in the well location download.

The next five digits are the permit number.  This is a sequential permit number issued by the Geological Survey.  Until 1998 the permit number was the only identification the Geological Survey used and for our purposes it is still pretty much all we need to identify a well.  In the example the permit number is 44190.

The last four digits are for different kicks of the same well; for example, horizontal drills, lost holes, or directional drills off the main hole.  Someone visiting the site would only see one well, but there may be several API numbers for that one well, all having different bottom hole locations.  The main number usually ends of four zeros.  There are special cases where the last four digits start with 7 (some brine disposal wells) 8 (some mineral wells) and 9 (wells drilled before permits were issued).  There are a few inconsistencies with the last four digits since Michigan didn't use API numbers until 1998 and they had to be retroactively assigned.

If looking at formation tops, it usually is only necessary to use the 0000 kick as any other kicks should pretty much contain the same data (or be null).