I'm not sure what you don't find clear in the Nexus Book. The short version is, a proxy repository is one that you are mirroring, and a hosted repository is one that you host on your server through the Nexus software. Hosted includes third party libraries that aren't public for some reason, see below.
6.2.1. Proxy Repository
A Proxy Repository is a proxy of a remote repository. By default, Nexus ships with the following configured proxy repositories:
Apache Snapshots
This repository contains snapshot releases from the Apache Software Foundation.
Codehaus Snapshots
This repository contains snapshot releases from Codehaus.
Central
This is the Central Repository containing release components. Formerly known as Maven Central, it is the default built-in repository for Apache Maven and directly supported in other build tools like Gradle, SBT or Ant/Ivy. Nexus connects to the Central Repository via HTTPS using the URL https://repo1.maven.org/maven2/.
6.2.2. Hosted Repository
A Hosted Repository is a repository that is hosted by Nexus. Nexus ships with the following configured hosted repositories:
3rd Party
This hosted repository should be used for third-party dependencies not available in the public Maven repositories. Examples of these dependencies could be commercial, proprietary libraries such as an Oracle JDBC driver that may be referenced by your organization.
Releases
This hosted repository is where your organization will publish internal releases.
Snapshots
This hosted repository is where your organization will publish internal snapshots.