When editing the field in the Structure tab, look for the "NULL" checkbox. When un-checked, this is the equivalent of the NOT NULL statement.

You must not have SQL_MODE set to strict on your installation.

Issue

SET SQL_MODE='STRICT_ALL_TABLES'

or add

SQL_MODE='STRICT_ALL_TABLES'

under [mysqld] into your my.cnf

To find my.cnf, look in the MySQL config file C:\xampp\mysql\bin\my.ini.

At the top of that file are some comments:

# You can copy this file to
# C:/xampp/mysql/bin/my.cnf to set global options,
# mysql-data-dir/my.cnf to set server-specific options (in this
# installation this directory is C:/xampp/mysql/data) or
# ~/.my.cnf to set user-specific options.

There it tells you where to find your my.cnf file.

Restart your mysql if necessary.

There must be a checkbox near input for setting NULL value. ( only on nullable columns )

( alternatievly, I prefer to use SQL language directly )

You just need to issue an ALTER TABLE statement like this:

ALTER TABLE table_name
MODIFY column_name [data type] NOT NULL,
MODIFY column_name_2 [data type] NOT NULL,
...
MODIFY column_name_x [data type] NOT NULL

Note that [data type] here is of course the data type of your column (i.e. VARCHAR(255) or whatever)

Just list all the columns you want to modify in this single statement.

If the column has no meaning in the NULL state, or should not be NULL because it's a required column, or required reference to another table, make it NOT NULL.

Otherwise, it doesn't really matter. Unless you need to store NULL values, it's somewhat arbitrary what you set.

What version of mysql is this?

What mode are you running in?

SELECT @@GLOBAL.SQL_MODE, @@SESSION.SQL_MODE;

(This should be run in the context of your application, just in case it is changing it).

MySQL is documented thus: http://dev.mysql.com/doc/refman/5.0/en/data-type-defaults.html

As of MySQL 5.0.2, if a column definition includes no explicit DEFAULT value, MySQL determines the default value as follows:

If the column can take NULL as a value, the column is defined with an explicit DEFAULT NULL clause. This is the same as before 5.0.2.

If the column cannot take NULL as the value, MySQL defines the column with no explicit DEFAULT clause. For data entry, if an INSERT or REPLACE statement includes no value for
the column, MySQL handles the column according to the SQL mode in effect at the time:

If strict SQL mode is not enabled, MySQL sets the column to the implicit default value for the column data type. 

My own testing fails to duplicate your issue

mysql> CREATE TABLE `my_table` (
    ->   `entry_id` int(11) NOT NULL AUTO_INCREMENT,
    ->   `address` varchar(512) NOT NULL,
    ->   `follow_up_to` int(11) DEFAULT NULL,
    ->   PRIMARY KEY (`entry_id`),
    ->   KEY `follow_up_to` (`follow_up_to`)
    -> ) ENGINE=InnoDB AUTO_INCREMENT=536 DEFAULT CHARSET=latin1;
Query OK, 0 rows affected, 2 warnings (0.16 sec)

mysql> INSERT INTO my_table VALUES (NULL, NULL, NULL);
ERROR 1048 (23000): Column 'address' cannot be null
mysql> INSERT INTO my_table (follow_up_to) VALUES (NULL);
Query OK, 1 row affected, 1 warning (0.10 sec)

mysql> SHOW WARNINGS;
+---------+------+----------------------------------------------+
| Level   | Code | Message                                      |
+---------+------+----------------------------------------------+
| Warning | 1364 | Field 'address' doesn't have a default value |
+---------+------+----------------------------------------------+
1 row in set (0.00 sec)

mysql> SELECT * FROM my_table;
+----------+---------+--------------+
| entry_id | address | follow_up_to |
+----------+---------+--------------+
|      537 |         |         NULL |
+----------+---------+--------------+
1 row in set (0.00 sec)

The fact the old data had nulls shouldn't matter. The Alter table should have 'truncated' the nulls into empty strings

mysql> SHOW CREATE TABLE my_table\G
*************************** 1. row ***************************
       Table: my_table
Create Table: CREATE TABLE `my_table` (
  `entry_id` int(11) NOT NULL AUTO_INCREMENT,
  `address` varchar(512) NOT NULL,
  `follow_up_to` int(11) DEFAULT NULL,
  PRIMARY KEY (`entry_id`),
  KEY `follow_up_to` (`follow_up_to`)
) ENGINE=InnoDB AUTO_INCREMENT=536 DEFAULT CHARSET=latin1
1 row in set (0.04 sec)

mysql> ALTER TABLE my_table MODIFY address VARCHAR(512) NULL DEFAULT NULL;
Query OK, 1 row affected (0.76 sec)
Records: 1  Duplicates: 0  Warnings: 0

mysql> INSERT INTO my_table VALUES (NULL, NULL, NULL), (NULL, NULL, NULL);
Query OK, 2 rows affected (0.05 sec)
Records: 2  Duplicates: 0  Warnings: 0

mysql> SELECT * FROM my_table;
+----------+---------+--------------+
| entry_id | address | follow_up_to |
+----------+---------+--------------+
|      535 |         |         NULL |
|      536 | NULL    |         NULL |
|      537 | NULL    |         NULL |
+----------+---------+--------------+
3 rows in set (0.04 sec)

mysql> ALTER TABLE my_table MODIFY address VARCHAR(512) NOT NULL;
Query OK, 3 rows affected, 2 warnings (0.83 sec)
Records: 3  Duplicates: 0  Warnings: 2

mysql> SHOW WARNINGS;
+---------+------+----------------------------------------------+
| Level   | Code | Message                                      |
+---------+------+----------------------------------------------+
| Warning | 1265 | Data truncated for column 'address' at row 2 |
| Warning | 1265 | Data truncated for column 'address' at row 3 |
+---------+------+----------------------------------------------+
2 rows in set (0.04 sec)

mysql> SELECT * FROM my_table;
+----------+---------+--------------+
| entry_id | address | follow_up_to |
+----------+---------+--------------+
|      535 |         |         NULL |
|      536 |         |         NULL |
|      537 |         |         NULL |
+----------+---------+--------------+
3 rows in set (0.04 sec)

mysql> INSERT INTO my_table VALUES (NULL, NULL, NULL), (NULL, NULL, NULL);
Query OK, 2 rows affected, 2 warnings (0.08 sec)
Records: 2  Duplicates: 0  Warnings: 2

mysql> SELECT * FROM my_table;
+----------+---------+--------------+
| entry_id | address | follow_up_to |
+----------+---------+--------------+
|      535 |         |         NULL |
|      536 |         |         NULL |
|      537 |         |         NULL |
|      538 |         |         NULL |
|      539 |         |         NULL |
+----------+---------+--------------+
5 rows in set (0.05 sec)

It actually doesn't accept NULL values it considers it as empty string. That's because you have your server in non-strict mode. That controls how MySQL handles invalid or missing values in inserts and updates. You can read more about modes here: http://dev.mysql.com/doc/refman/5.6/en/sql-mode.html#sql-mode-strict

mysql> insert into cities(state_id) values (20);
Query OK, 1 row affected, 1 warning (0.07 sec)

mysql> show warnings;
+---------+------+-------------------------------------------+
| Level   | Code | Message                                   |
+---------+------+-------------------------------------------+
| Warning | 1364 | Field 'name' doesn't have a default value |
+---------+------+-------------------------------------------+

mysql> select name is null from cities where id = LAST_INSERT_ID();
+--------------+
| name is null |
+--------------+
|            0 |
+--------------+
1 row in set (0.00 sec)

mysql> SET sql_mode = 'STRICT_ALL_TABLES';
Query OK, 0 rows affected (0.00 sec)

mysql> insert into cities(state_id) values (20);
ERROR 1364 (HY000): Field 'name' doesn't have a default value

You want the following:

ALTER TABLE mytable MODIFY mycolumn VARCHAR(255);

Columns are nullable by default. As long as the column is not declared UNIQUE or NOT NULL, there shouldn't be any problems.