The equivalent of
df %>% groupby(col1) %>% summarize(col2_agg=max(col2), col3_agg=min(col3))
is
df.groupby('col1').agg({'col2': 'max', 'col3': 'min'})
which returns 
      col2  col3
col1            
1        5    -5
2        9    -9
The returning object is a pandas.DataFrame with an index called col1 and columns named col2 and col3. By default, when you group your data pandas sets the grouping column(s) as index for efficient access and modification. However, if you don't want that, there are two alternatives to set col1 as a column.
- Pass - as_index=False:
 - df.groupby('col1', as_index=False).agg({'col2': 'max', 'col3': 'min'})
 
- Call - reset_index:
 - df.groupby('col1').agg({'col2': 'max', 'col3': 'min'}).reset_index()
 
both yield
col1  col2  col3           
   1     5    -5
   2     9    -9
You can also pass multiple functions to groupby.agg.
agg_df = df.groupby('col1').agg({'col2': ['max', 'min', 'std'], 
                                 'col3': ['size', 'std', 'mean', 'max']})
This also returns a DataFrame but now it has a MultiIndex for columns.
     col2               col3                   
      max min       std size       std mean max
col1                                           
1       5   1  1.581139    5  1.581139   -3  -1
2       9   0  3.535534    5  3.535534   -6   0
MultiIndex is very handy for selection and grouping. Here are some examples:
agg_df['col2']  # select the second column
      max  min       std
col1                    
1       5    1  1.581139
2       9    0  3.535534
agg_df[('col2', 'max')]  # select the maximum of the second column
Out: 
col1
1    5
2    9
Name: (col2, max), dtype: int64
agg_df.xs('max', axis=1, level=1)  # select the maximum of all columns
Out: 
      col2  col3
col1            
1        5    -1
2        9     0
Earlier (before version 0.20.0) it was possible to use dictionaries for renaming the columns in the agg call. For example
df.groupby('col1')['col2'].agg({'max_col2': 'max'})
would return the maximum of the second column as max_col2:
      max_col2
col1          
1            5
2            9
However, it was deprecated in favor of the rename method:
df.groupby('col1')['col2'].agg(['max']).rename(columns={'max': 'col2_max'})
      col2_max
col1          
1            5
2            9
It can get verbose for a DataFrame like agg_df defined above. You can use a renaming function to flatten those levels in that case:
agg_df.columns = ['_'.join(col) for col in agg_df.columns]
      col2_max  col2_min  col2_std  col3_size  col3_std  col3_mean  col3_max
col1                                                                        
1            5         1  1.581139          5  1.581139         -3        -1
2            9         0  3.535534          5  3.535534         -6         0
For operations like groupby().summarize(newcolumn=max(col2 * col3)), you can still use agg by first adding a new column with assign.
df.assign(new_col=df.eval('col2 * col3')).groupby('col1').agg('max') 
      col2  col3  new_col
col1                     
1        5    -1       -1
2        9     0        0
This returns maximum for old and new columns but as always you can slice that.
df.assign(new_col=df.eval('col2 * col3')).groupby('col1')['new_col'].agg('max')
col1
1   -1
2    0
Name: new_col, dtype: int64
With groupby.apply this would be shorter:
df.groupby('col1').apply(lambda x: (x.col2 * x.col3).max())
col1
1   -1
2    0
dtype: int64
However, groupby.apply treats this as a custom function so it is not vectorized. Up to now, the functions we passed to agg ('min', 'max', 'min', 'size' etc.) are vectorized and these are aliases for those optimized functions. You can replace df.groupby('col1').agg('min') with df.groupby('col1').agg(min), df.groupby('col1').agg(np.min) or df.groupby('col1').min() and they will all execute the same function. You will not see the same efficiency when you use custom functions.
Lastly, as of version 0.20, agg can be used on DataFrames directly, without having to group first. See examples here.