Consider the following example:
I have a dataset of Movielens-
u.item.csv
ID|MOVIE NAME (YEAR)|REL.DATE|NULL|IMDB LINK|A|B|C|D|E         
        
I think you need map by Series created by set_index:
print (df1.set_index('ID')['MOVIE NAME (YEAR)'])
ID
1     Toy Story (1995)
2     GoldenEye (1995)
3    Four Rooms (1995)
Name: MOVIE NAME (YEAR), dtype: object
df2['movie_id'] = df2['movie_id'].map(df1.set_index('ID')['MOVIE NAME (YEAR)'])
print (df2)
   user_id           movie_id  rating  unix_timestamp
0        1   Toy Story (1995)       5       874965758
1        1   GoldenEye (1995)       3       876893171
2        1  Four Rooms (1995)       4       878542960
Or use replace:
df2['movie_id'] = df2['movie_id'].replace(df1.set_index('ID')['MOVIE NAME (YEAR)'])
print (df2)
   user_id           movie_id  rating  unix_timestamp
0        1   Toy Story (1995)       5       874965758
1        1   GoldenEye (1995)       3       876893171
2        1  Four Rooms (1995)       4       878542960
Difference is if not match, map create NaN and replace let original value:
print (df2)
   user_id  movie_id  rating  unix_timestamp
0        1         1       5       874965758
1        1         2       3       876893171
2        1         5       4       878542960 <- 5 not match
df2['movie_id'] = df2['movie_id'].map(df1.set_index('ID')['MOVIE NAME (YEAR)'])
print (df2)
   user_id          movie_id  rating  unix_timestamp
0        1  Toy Story (1995)       5       874965758
1        1  GoldenEye (1995)       3       876893171
2        1               NaN       4       878542960
df2['movie_id'] = df2['movie_id'].replace(df1.set_index('ID')['MOVIE NAME (YEAR)'])
print (df2)
   user_id          movie_id  rating  unix_timestamp
0        1  Toy Story (1995)       5       874965758
1        1  GoldenEye (1995)       3       876893171
2        1                 5       4       878542960