Please notice the difference of copying at the file level and cloning partitions or whole drives at the device level.
Copy files as described by muru
In this case you use the mounted partitions, where you see the directories and files.
rsync is a very good tool for backup and synchronizing.
Clone devices as described below
Cloning devices is very risky, and you must check and double-check, that you write to the correct device. Otherwise you might overwrite your family pictures ... and it is only a minor typing error away. When cloning you should unmount all partitions on the source and target devices. You already noticed that you should clone to a device of at least the same same.
sudo dd if=/dev/sdx of=/dev/sdy bs=4096
clones a device for example USB pendrive (from drive x to drive y).
sudo dd if=/dev/sdxm of=/dev/sdyn bs=4096
clones a partition to another partition (from partition xm to partition yn, where x and y are drive letters and m and n are partition numbers).
sudo dd if=/dev/sdx of=file.img bs=4096
creates an image file of a drive.
sudo dd if=/dev/sdx bs=4096 | xz > file.img.xz
creates a compressed image file.
sudo if=ubuntu.iso of=/dev/sdx bs=4096
creates a USB boot drive from an Ubuntu iso file.
-o-
It is possible to clone with cp instead of dd and get similar results.
Tools
In order to reduce the risk and to make the process easier for you, there are tools to help with this process, particularly to create boot drives from iso files and [compressed] image filas. Examples: the Ubuntu Startup Disk Creator (in 16.04 LTS and newer), Disks (gnome-disks), mkusb.
Clonezilla is a tools that can select the used blocks on the 'drive surface' and copy/clone them, while it skips the unused blocks. This makes cloning with Clonezilla much faster than cloning with dd (or cp), particularly, in a drive with a lot of free space.
cp
usually works best for copying files, as then you don't accidentally nuke the destination partition's filesystem (n.b. it is possible to have more than one partition on a disk). For recovery (when the filesystem can't be recognised or returns errors as it formatted incorrectly or the disk is not functioning correctly) in general you need to try to use the disk as little as possible, and carry out recovery operations on a copy - this is covered in @muru 's linked question, though I think there are others around this site too... – Wilf Dec 29 '16 at 16:54