I'm having a little trouble figuring out exactly what your tables would appear to be, but as I understand you, table1 has data in column a that you want to move to table2 column b and it sounds to me like a column 'ID' is the relationship between the two?
If that is the case, using Phil's suggestion, it would be someting like
update table2
set B = table1.A
from table2
inner join table1
on table1.ID = table2.ID
If I have misunderstood you, apologies.