Find and compress file in Python script
First thing's first, instead of calling out to the gzip program with os.system, use the gzip module in Python.  This will almost always be more portable: http://docs.python.org/2/library/gzip.html

Second of all, instead of using os.path.getsize and os.path.getmtime just use a single os.stat call: http://docs.python.org/2/library/os.html#os.stat  You can call this once and cache the results for a single file.

Don't require calling self.date() in order to set the unix_time and file_date attributes.  Instead make those into properties.  For example you could write that like:

class CheckFile(object):
    'Operation on file like get size,file time etc..'

    def __init__(self, fname):
        self.fname = fname
        self._stat = None

    @property
    def stat(self):
        if self._stat is None:
            # Note: This will not update automatically if your file's stats change
            self._stat = os.stat(self.fname)
        return self._stat

    @property
    def unix_time(self):
        return self.stat.st_mtime

    @property
    def datetime(self):
        return datetime.fromtimestamp(self.unix_time)

    @property
    def age_in_days(self):
        return (datetime.today() - self.datetime).days


All that said, although I wouldn't discourage anyone from wanting to learn object-oriented programming, for this particular application writing a class is overkill :)