Not a vallid month

also
    date between to_char(date_of,'dd.mm.yyyy') and to_char(date_to,'dd.mm.yyyy');
seems extremely odd too.  What is the datatype of the 'date' column involved?

for any date data types, please use only DATE data type so that you will not get into these type of issues invariably though you can play with the dates by storing them in varchar2.

Even if i pass everything as date for example:

create or replace
PROCEDURE   p_run_by_param
IS
begin
DECLARE

	date_of date; 
	date_to date; 

begin


  date_of := trunc(SYSDATE,'mm');
  date_to := SYSDATE;
  
    p_real(date_of,date_to);
 
 end;

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and

create or replace
PROCEDURE p_real
(date_of date,date_to date)
is
BEGIN
    
   select
   *
   from table
   where date between date_of and date_to;
  
 end;

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it still fails with same error.

But again : It WORKS is SQL developer ... fails only when i create job.

Are you sure there is no other date conversion, implied or explicit, occurring elsewhere in the code?  That sort of error is during a to_date() type of operation.  
The difference between SQL Developer and a job ... will most likely by SESSION based environment defaults (particularly NLS_DATE_FORMAT) - I am guessing that SQL Developer is setting one on login which is different to the system default used by the job.
This is why implicit conversions are bad, very bad.

hmm its quite strange really.   I even fail wih this: (note i have commented call to procedure)

create or replace
PROCEDURE   p_run_by_param
IS
begin
DECLARE

	date_of date; 
	date_to date; 

begin


  date_of := trunc(SYSDATE,'mm');
  date_to := SYSDATE;
  
   -- p_real(date_of,date_to);
 
 end;

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have also tried this and same error.  What would be correct form of this procedure?

create or replace
PROCEDURE   p_run_by_param
IS
begin
DECLARE

	date_of date; 
	date_to date; 

begin


  date_of := '10.08.2012';
  date_to := 12.09.2012;
  
   -- p_real(date_of,date_to);
 
 end;

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Procedure 2 will not compile because you cannot select something in a procedure without assigning the result somewhere. So you would have to declare a variable and select the result into that, something like this:

create or replace PROCEDURE p_real (date_of date,date_to date) as
date_count number;
begin
select count (*) into date_count from table where date between date_of and date_to;
end;

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What is the exact error message and how are you running the job?

I create job inside of enterprize manager:

BEGIN
sys.dbms_scheduler.create_job( 
job_name => '"SYSTEM"."p_real"',
job_type => 'PLSQL_BLOCK',
job_action => 'scott.p_real;',
repeat_interval => 'FREQ=DAILY;BYHOUR=10;BYMINUTE=10',
start_date => systimestamp at time zone 'Europe/Vienna',
job_class => 'DEFAULT_JOB_CLASS',
comments => 'test',
auto_drop => FALSE,
enabled => TRUE);
END;

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and exact error msg is:

ORA-01843: not a valid month

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That job schedule is calling p_real without parameters, not p_run_by_param

@gerwinjansen:

select systimestamp at time zone 'Europe/Vienna' from dual;

Result does not look like a date to me... I'd remove the at time zone part.

This was the solution!  Thank you!