How can I write some information inside a photo file like jpg or gif without destroying the image? and of course without showing it on the photo since the whole idea is to send information in the file of photo undetected by anyone (to provide security/privacy to some extent)!
You can store some information in image metadata. In fact that's how man digital cameras 'tag' the photos their making (camera model, date and time, GPS coords etc.).
This data format is called EXIF (Exchangeable Image File Format). There are a lot of examples how to use it in programming languages. Here's the example in Java.
If you want to prevent users from reading this data you can encrypt them somehow, but they will always be able to remove it from your picture (by opening in Photoshop and using Save As for example).