UPDATE: installation without root privileges below

I advise you to not install packages manually on ubuntu system if there is already a (semi-official) repository able to solve your problem. Further, use Oracle JDK for development, just to avoid (very sporadic) compatibility issues (i've tried many years ago, it's surely better now).

Add the webupd8 repo to your system:

Copysudo add-apt-repository ppa:webupd8team/java
sudo apt-get update

Install your preferred version of jdk (versions from java-6 to java-9 available):

Copysudo apt-get install oracle-java8-installer

You can also install multiple version of jdk, mixing openjdk and oracle versions. Then you can use the command update-java-alternatives to switch between installed version:

Copy# list available jdk
update-java-alternatives --list

# use jdk7
sudo update-java-alternatives --set java-7-oracle

# use jdk8
sudo update-java-alternatives --set java-8-oracle

Requirements

If you get add-apt-repository: command not found be sure to have software-properties-common installed:

Copysudo apt-get install software-properties-common

If you're using an older version Ubuntu:

Copysudo apt-get install python-software-properties

JDK installation without root privileges

If you haven't administrator rights on your target machine your simplest bet is to use sdkman to install the zulu certified openjdk:

Copycurl -s "https://get.sdkman.io" | bash
source "$HOME/.sdkman/bin/sdkman-init.sh"
sdk install java

NOTE: sdkman allow to install also the official Oracle JDK, although it's not a the default option. View available versions with:

Copysdk ls java

Install the chosen version with:

Copysdk install java <version>

For example:

Copysdk install java 9.0.1-oracle

Glossary of commands

• sudo <command> [command_arguments]: execute a command with the superuser privilege.

• add-apt-repository <PPA_id>: Ubuntu (just like every Debian derivatives and generally speaking every Linux distribution) has a main repository of packages that handle things like package dependencies and updating. In Ubuntu is possible to extend the main repository using a PPA (Personal Package Archive) that usually contains packages not available in the system (just like oracle jdk) or updated versions of available ones (example: LibreOffice 5 in LTS is available only through this PPA).

• apt-get [install|update|upgrade|purge|...]: it's "the" command-line package handler used to manipulate the state of every repository on the system (installing / updating / upgrading can be viewed as an alteration of the repository current state).

In our case: with the command sudo add-apt-repository ppa:webupd8team/java we inform the system that the next repository update must retrieve packages information also from webupd8 repo.

With sudo apt-get update we actually update the system repository (all this operations requires superuser privileges, so we prepend sudo to the commands).

sudo apt-get install oracle-java8-installer

• update-java-alternatives (a specific java version of update-alternatives): in Ubuntu several packages provides the same functionality (browse the internet, compile mails, edit a text file or provides java/javac executables...). To allows the system to choose the user favourites tool given a specific task a mechanism using symlinks under /etc/alternatives/ is used. Try to update the jdk as indicated above (switch between java 7 and java 8) and view how change the output of this command:

  ls -l /etc/alternatives/java*

In our case: sudo update-java-alternatives --set java-8-oracle update symlinks under /etc/alternatives to point to java-8-oracle executables.

Extras:

• man <command>: start using man to read a really well written and detailed help on (almost) every shell command and its options (every command i mention in this little answer has a man page, try man update-java-alternatives).

• apt-cache search <search_key>: query the APT cache to search for a package related with the search_key provided (can be the package name or some word in package description).

• apt-cache show <package>: provides APT information for a specific package (package version, installed or not, description).

You can download and install it in /opt/ so it does not interfere with the currently installed version. Commands:

Download:

wget https://download.java.net/java/GA/jdk18/43f95e8614114aeaa8e8a5fcf20a682d/36/GPL/openjdk-18_linux-x64_bin.tar.gz

Extract the download (replace {version} with the one downloaded):

tar -xvf openjdk-18._linux-{version}

Move it to /opt/:

sudo mv jdk-18* /opt/

(replace jdk-18 to something else if you want) and set the environment variables:

export JAVA_HOME=/opt/jdk-18
export PATH=$PATH:$JAVA_HOME/bin

To check if that went correctly:

cd /opt/jdk-18/
java --version
echo $JAVA_HOME

By re-doing the exports to exclude /opt/jdk-18 you restore the old situation

[17-08-2022] thank to @nobody (see comment to the question)

For all older version use this link:

https://launchpad.net/~openjdk-r/+archive/ubuntu/ppa

sudo add-apt-repository ppa:openjdk-r/ppa
sudo apt update

and select your version:

sudo apt install openjdk-15-jdk

apt can’t locate openjdk-16-jre-headless because the OpenJDK 16 packages aren’t available in any Debian release.

The only release available to you in Debian 10 is OpenJDK 11. Debian 11 has OpenJDK 17. Both of these are releases with long-term support.

Running

Copysudo apt update

before trying to install a second JDK (g00se suggested I had to) helped. It was ChatGPT's advice. If you need to know where the JDK is installed, try (also GPT's idea)

Copyupdate-alternatives --list java

With that information, you should be able to register that JAVA_HOME thing. In my case, it was

Copyexport JAVA_HOME=/usr/lib/jvm/java-17-openjdk-amd64

I hope it helps somebody one day

Command Line option - Ubuntu

Copysudo apt-get install python-software-properties
sudo add-apt-repository ppa:webupd8team/java
sudo apt-get update

Then in terminal

Copysudo apt-get install oracle-java8-installer

When there are multiple Java installations on your System, the Java version to use as default can be chosen. To do this, execute the following command.

Copysudo update-alternatives --config java
sudo update-alternatives --config javac
sudo update-alternatives --config javaws

Edit - Manual Java Installation

Download oracle jdk

Copyhttp://www.oracle.com/technetwork/java/javase/downloads/index.html

Extract zip into desired folder

Copy e.g  /usr/local/  after extract /usr/local/jdk1.8.0_65

Setup

Copysudo update-alternatives --install  /usr/bin/java java /usr/local/jdk1.8.0_65/bin/java 1
sudo update-alternatives --install  /usr/bin/javac javac /usr/local/jdk1.8.0_65/bin/javac 1
sudo update-alternatives --install  /usr/bin/javaws javaws /usr/local/jdk1.8.0_65/bin/javaws 1

sudo update-alternatives --set  java /usr/local/jdk1.8.0_65/bin/java
sudo update-alternatives --set  javac /usr/local/jdk1.8.0_65/bin/javac
sudo update-alternatives --set  javaws /usr/local/jdk1.8.0_65/bin/javaws

Edit /etc/environment set JAVA_HOME path for external applications like Eclipse and Idea

For Ubuntu 22.04, you should be able to simply:
sudo apt install openjdk-21-jdk

It's in the Universe repository, so be sure you have that enabled.

It wasn't available yet when you asked the question, but it is now.

If you add a PPA or other non-Ubuntu version of OpenJDK, remember that it's likely to be incompatible with the (supported Ubuntu repository) Ubuntu package. So keep track of what you install; you might need to remove all of it.

Try a quick apt-get update to get the latest package lists from repos, and then apt-cache search openjdk to check out what the latest offered version is that's available to you, if any are.

Most likely if this is a fresh base install you're going to have to go into your /etc/apt/sources.list and make sure you have the non-free package lists enabled, which aren't enabled by default in Debian I believe. So check this How do I enable non-free packages on Debian?