JRE - Java Runtime Environment. It comes with the Java (
executable for running Java application. This is typical for end users
who use it to run compiled Java applications (in jar file distribution),
and not doing any development. JRE is smaller to download than JDK.
JDK - Java Development Kit. It comes with Java compiler (
many other development tools; and it also include the JRE as well. This
is for Java development and running Java applications together. As a
developer, you usually want to get this as single package.
Oracle sponsors and drives the official open source OpenJDK project, where new Java edition are develope. It provides binary releases that developers may download for various OS envs. (Note starting JDK9, you may download binary package that simply untar/unzip into a directory without need of installer!)
Oracle also provides commercial version of the Java SE JDK. These JDK packages has different licenses for usage, and are built with different installer format. Users usually require local Admin Rights to install it.