How can a script check if it's being run as root?

#!/bin/bash
if [[ $EUID -ne 0 ]]; then
   echo "This script must be run as root" 
   exit 1
fi

A root user does not have to be named "root". whoami returns the first username with user ID 0. $USER contains the name of the logged in user, which can have user ID 0, but have a different name.

The only reliable program to check whether the account is logged in as root, or not:

id -u

I use -u for the effective user ID, not -r for the real user ID. Permissions are determined by the effective user ID, not the real one.

Tests

/etc/passwd contains the following usernames with user ID 0 in the given order:

rootx
root2

Logged in as root2, gives the next results:

• whoami: rootx
• echo $USER: root2 (this returns an empty string if the program was started in an empty environment, e.g. env -i sh -c 'echo $USER')
• id -u: 0 As you can see, the other programs failed in this check, only id -u passed.

The updated script would looks like this:

#!/bin/bash
if ! [ $(id -u) = 0 ]; then
   echo "I am not root!"
   exit 1
fi

As @Lekensteyn said you should use effective user ID. You don't need to call id -u in bash:

#!/bin/bash

if [[ $EUID -ne 0 ]]; then
   echo "You must be root to do this." 1>&2
   exit 100
fi

@geirha's suggestion from the comments uses arithmetic evaluation:

#!/bin/bash

if (( EUID != 0 )); then
   echo "You must be root to do this." 1>&2
   exit 100
fi