There sure is a lot of terminology in the Java world and that can create a significant learning curve for new developers. It's not that it's particularly difficult to pass JSON or XML documents around using Java, it's just that the various bits and pieces you need to do it have sprouted terminology over the years. I've tried to list my understanding of the terms you've used below...
XML - you know what XML is, right? The extensible markup language. It's what we had before JSON became the big thing.
JSON - oh, well, JSON is the new big thing. It's a human readable object serialisation format, less verbose than XML. Very popular nowadays. It's the new magic bullet, good for what ails ya, gonna solve all your problems...
JAXB - the "Java Architecture for XML Binding" in the Java ecosystem is the primary mechanism for turning XML data into objects which you can then interact with, and vice versa. It's important to realise that it's an API and not an implementation, so it mostly defines a set of annotations and simple classes / interfaces in the javax.xml.bind package. To do anything useful with JAXB you need an implementation. There's a reference implementation included in the Glassfish application server. Most application servers will have some kind of implementation of JAXB.
Jackson - a library for data binding. It supports both XML and JSON as document formats, and implements the JAXB API. You can use Jackson as your implementation of JAXB, or you can just use the Jackson API directly.
EclipseLink Moxy - an alternative implementation of the JAXB API. Like Jackson, it also has its own API. You can choose to use it, or not. You probably don't want to use both Jackson and Moxy.
Jersey-media-moxy - as you mentioned, Jersey is an implementation of JAX-RS. One aspect of JAX-RS is passing documents around - often XML or JSON. To do that Jersey needs to know what underlying library to use for data-binding or stream processing. So jersey-media-moxy exists as a kind of jersey plugin dependency which you can use to configure Jersey to use Moxy for your object serialisation needs. There's an equivalent package for using jackson called jersey-media-json-jackson.
Jettison - Yet Another serialisation library for converting Java objects to Json and back.
JSON-P - an API for processing JSON either as a stream of events or via data-binding to an object. This API is still in development. You might ask how it is that anybody does json processing without it - the answer is that they either utilise proprietary library APIs (such as Jackson or Moxy) or they use a library which repurposes the JAXB API to work with JSON (Jackson definitely allows this, I'm not sure about Moxy). JSON-P will make it easier to work directly with JSON features, without all the XML-concepts which JAXB brings in.
