How do I apply a value from a dataframe based on the value of a multi-index of another dataframe?

what I suggest you do is 1) rename the columns of Dataframe 2 to the respective names of the index columns of Dataframe 1, 2) reset the index on Dataframe 1, and 3) merge the two tables based on the now matching column names. Afterwards you can compute whatever you like. The MultiIndex on the columns of Dataframe 2 adds a bit of additional overhead.

Explicitly:

import pandas as pd

# re-create table1
row_index = pd.MultiIndex.from_tuples([(128, 12345, 'AAA')])
row_index.names=['Index_A', 'Index_B', 'Index_C']
table1 = pd.DataFrame(data={'mean': 123, 'std': 2, 'count': 4}, index=row_index)
table1.columns = pd.MultiIndex.from_tuples(zip(['Assay A'] * 3, table1.columns))
print "*** table 1:"
print table1
print ""

# re-create table2
table2 = pd.DataFrame([{'Col_A': 128, 'Col_B': 12345, 'Col_C': 'AAA', 'mean': 456}], index=[1])
table2.index.name = 'Index'
print "*** table 2:"
print table2
print ""

# re-name columns of table2 to match names of respective index columns in table1
table2 = table2.rename(columns={'Col_A': 'Index_A', 'Col_B': 'Index_B', 'Col_C': 'Index_C'})

# Drop 'Assay A' index level on columns of table1;
# without doing that, the following reset_index() will produce a column multi-index
# for Index_A/B/C, so column names will not match the simple column index of table2_renamed.
# If you need to keep the 'Assay A' level here, you will need to also construct a column
# multi-index for table2_renamed (with empty values for the second level).
table1.columns = table1.columns.levels[1]

# Move index columns of table1 back to regular columns
table1 = table1.reset_index()

# Merge the two tables on the now common column names. 'mean' appears in both tables,
# give the column from table2 a suffix '_2'.
joint = pd.merge(table1.reset_index(), table2, on=['Index_A', 'Index_B', 'Index_C'], suffixes={'', '_2'})
print "*** joint, before re-setting index:"
print joint
print ""

# Restore index of the joint table
joint = joint.set_index(['Index_A', 'Index_B', 'Index_C'])

# Compute the 'result'
joint['result'] = joint['mean'] / joint['mean_2']

# drop unused columns
joint = joint.drop(['index', 'mean_2'], axis=1)

# restore column index level
joint.columns = pd.MultiIndex.from_tuples(zip(['Assay A'] * 4, joint.columns))

print "*** final result:"
print joint
print ""

The script output is:

*** table 1:
                        Assay A         
                          count mean std
Index_A Index_B Index_C                 
128     12345   AAA           4  123   2

*** table 2:
       Col_A  Col_B Col_C  mean
Index                          
1        128  12345   AAA   456

*** joint, before re-setting index:
   index  Index_A  Index_B Index_C  count  mean  std  mean_2
0      0      128    12345     AAA      4   123    2     456

*** final result:
                        Assay A                   
                          count mean std    result
Index_A Index_B Index_C                           
128     12345   AAA           4  123   2  0.269737

Hope that helps!