I have a pandas dataframe in which one column of text strings contains comma-separated values. I want to split each CSV field and create a new row per entry (as         
        
Based on the excellent @DMulligan's solution, here is a generic vectorized (no loops) function which splits a column of a dataframe into multiple rows, and merges it back to the original dataframe. It also uses a great generic change_column_order function from this answer.
def change_column_order(df, col_name, index):
    cols = df.columns.tolist()
    cols.remove(col_name)
    cols.insert(index, col_name)
    return df[cols]
def split_df(dataframe, col_name, sep):
    orig_col_index = dataframe.columns.tolist().index(col_name)
    orig_index_name = dataframe.index.name
    orig_columns = dataframe.columns
    dataframe = dataframe.reset_index()  # we need a natural 0-based index for proper merge
    index_col_name = (set(dataframe.columns) - set(orig_columns)).pop()
    df_split = pd.DataFrame(
        pd.DataFrame(dataframe[col_name].str.split(sep).tolist())
        .stack().reset_index(level=1, drop=1), columns=[col_name])
    df = dataframe.drop(col_name, axis=1)
    df = pd.merge(df, df_split, left_index=True, right_index=True, how='inner')
    df = df.set_index(index_col_name)
    df.index.name = orig_index_name
    # merge adds the column to the last place, so we need to move it back
    return change_column_order(df, col_name, orig_col_index)
Example:
df = pd.DataFrame([['a:b', 1, 4], ['c:d', 2, 5], ['e:f:g:h', 3, 6]], 
                  columns=['Name', 'A', 'B'], index=[10, 12, 13])
df
        Name    A   B
    10   a:b     1   4
    12   c:d     2   5
    13   e:f:g:h 3   6
split_df(df, 'Name', ':')
    Name    A   B
10   a       1   4
10   b       1   4
12   c       2   5
12   d       2   5
13   e       3   6
13   f       3   6    
13   g       3   6    
13   h       3   6    
Note that it preserves the original index and order of the columns. It also works with dataframes which have non-sequential index.
I have come up with the following solution to this problem:
def iter_var1(d):
    for _, row in d.iterrows():
        for v in row["var1"].split(","):
            yield (v, row["var2"])
new_a = DataFrame.from_records([i for i in iter_var1(a)],
        columns=["var1", "var2"])
Series and DataFrame methods define a .explode() method that explodes lists into separate rows. See the docs section on Exploding a list-like column.
Since you have a list of comma separated strings, split the string on comma to get a list of elements, then call explode on that column.
df = pd.DataFrame({'var1': ['a,b,c', 'd,e,f'], 'var2': [1, 2]})
df
    var1  var2
0  a,b,c     1
1  d,e,f     2
df.assign(var1=df['var1'].str.split(',')).explode('var1')
  var1  var2
0    a     1
0    b     1
0    c     1
1    d     2
1    e     2
1    f     2
Note that explode only works on a single column (for now).
NaNs and empty lists get the treatment they deserve without you having to jump through hoops to get it right.
df = pd.DataFrame({'var1': ['d,e,f', '', np.nan], 'var2': [1, 2, 3]})
df
    var1  var2
0  d,e,f     1
1            2
2    NaN     3
df['var1'].str.split(',')
0    [d, e, f]
1           []
2          NaN
df.assign(var1=df['var1'].str.split(',')).explode('var1')
  var1  var2
0    d     1
0    e     1
0    f     1
1          2  # empty list entry becomes empty string after exploding 
2  NaN     3  # NaN left un-touched
This is a serious advantage over ravel + repeat -based solutions (which ignore empty lists completely, and choke on NaNs).
How about something like this:
In [55]: pd.concat([Series(row['var2'], row['var1'].split(','))              
                    for _, row in a.iterrows()]).reset_index()
Out[55]: 
  index  0
0     a  1
1     b  1
2     c  1
3     d  2
4     e  2
5     f  2
Then you just have to rename the columns
I came up with a solution for dataframes with arbitrary numbers of columns (while still only separating one column's entries at a time).
def splitDataFrameList(df,target_column,separator):
    ''' df = dataframe to split,
    target_column = the column containing the values to split
    separator = the symbol used to perform the split
    returns: a dataframe with each entry for the target column separated, with each element moved into a new row. 
    The values in the other columns are duplicated across the newly divided rows.
    '''
    def splitListToRows(row,row_accumulator,target_column,separator):
        split_row = row[target_column].split(separator)
        for s in split_row:
            new_row = row.to_dict()
            new_row[target_column] = s
            row_accumulator.append(new_row)
    new_rows = []
    df.apply(splitListToRows,axis=1,args = (new_rows,target_column,separator))
    new_df = pandas.DataFrame(new_rows)
    return new_df
Here's a function I wrote for this common task. It's more efficient than the Series/stack methods. Column order and names are retained.
def tidy_split(df, column, sep='|', keep=False):
    """
    Split the values of a column and expand so the new DataFrame has one split
    value per row. Filters rows where the column is missing.
    Params
    ------
    df : pandas.DataFrame
        dataframe with the column to split and expand
    column : str
        the column to split and expand
    sep : str
        the string used to split the column's values
    keep : bool
        whether to retain the presplit value as it's own row
    Returns
    -------
    pandas.DataFrame
        Returns a dataframe with the same columns as `df`.
    """
    indexes = list()
    new_values = list()
    df = df.dropna(subset=[column])
    for i, presplit in enumerate(df[column].astype(str)):
        values = presplit.split(sep)
        if keep and len(values) > 1:
            indexes.append(i)
            new_values.append(presplit)
        for value in values:
            indexes.append(i)
            new_values.append(value)
    new_df = df.iloc[indexes, :].copy()
    new_df[column] = new_values
    return new_df
With this function, the original question is as simple as:
tidy_split(a, 'var1', sep=',')