Why use secure deletion?
Operating systems do not actually remove the contents of a file when it is deleted, even after emptying the trash or explicitly removing the file, from the command line for example.
Instead, they simply remove the file's entry from the file system directory, because this requires less work and is therefore faster. The contents of the file—the actual data—remain on the storage medium. The data will remain there until the operating system reuses the space for new data.
Likewise, reformatting, repartitioning or reimaging a system is not always guaranteed to write to every area of the disk, though all will cause the disk to appear empty or, in the case of reimaging, empty except for the files present in the image, to most software.
Finally, even when the storage medium is overwritten, physical properties of the medium may make it possible to recover the previous contents. In most cases however, this recovery is not possible by just reading from the storage device in the usual way, but requires using laboratory techniques such as disassembling the device and directly accessing/reading from its components.
Quoted from Wikipedia: Secure file deletion.
Warning about USB sticks and solid-state drives
The methods described below will not work as expected on USB sticks and solid-state drives.
- The existing hard drive-oriented techniques for secure deletion of individual files are not effective.
- Overwriting twice the entire drive is usually, but not always, sufficient to securely clean the drive.
For more details read, the corresponding section of the Wikipedia article on Secure file deletion.
Securely delete files
In Tails you can securely delete files thanks to an extension of the Nautilus file manager called Nautilus Wipe.
Select the files you want to securely delete
Open Nautilus, either from the Places menu or the Computer icon on the desktop.
Navigate to the folder containing the files that you want to delete.
Select the files that you want to delete with the mouse.

Securely delete them using Nautilus Wipe
Securely delete them by doing right-click ▸ Wipe.

Confirm.

The deletion will start. It can last from a few seconds to several minutes, according to the size of the files. Be patient…

Once the deletion will be done you should be prompted with a message saying:

Securely clean available disk space
In order to clean up the contents of all files that were previously suppressed but not securely deleted from a disk, it is also possible to securely clean all the free space on the disk.
The disk or the folder may or may not contain other files. Those files will not be deleted during the operation.
Navigate to the disk you want to securely clean
Open Nautilus, either from the Places menu or the Computer icon on the desktop.
Navigate to a folder on the disk that you want to clean.
Securely clean the available space using Nautilus Wipe
Securely clean the available diskspace by doing right-click ▸ Wipe available diskspace.

Confirm.

The cleaning will start. It can last from a few minutes to a few hours, according to the size of the available diskspace. Be patient…
Note that a file called oooooooo.ooo is created in the folder. Nautilus
Wipe will try to make it as big as possible to use all the available
diskspace and then will securely delete it.

Once the cleaning will be done you should be prompted with a message saying:

