I don't know what this is called but I have two worksheets, column A
in worksheet a is a subset of column A in worksheet b. I want to find
all of the values in common in column A and move the row with common A
values from worksheet b to that matching row in worksheet a. Or end
up with a third worksheet with this effect. Any help?

To get started, you might consider something like this...

Create a new worksheet. In A1 of the new worksheet, put (all in one line):
=IF(COUNTIF(b!$A:$A,a!$A1)0,
VLOOKUP(a!$A1,b!$A:$Z,COLUMN(),FALSE),a!A1)

Select A1 and use
Edit Copy

Select the entire range of data on the new sheet and use
Edit Paste

Modify to suit.