What it does
Maven is a "build management tool", it is for defining how your .java files get compiled to .class, packaged into .jar (or .war or .ear) files, (pre/post)processed with tools, managing your CLASSPATH, and all others sorts of tasks that are required to build your project. It is similar to Apache Ant or Gradle or Makefiles in C/C++, but it attempts to be completely self-contained in it that you shouldn't need any additional tools or scripts by incorporating other common tasks like downloading & installing necessary libraries etc.
It is also designed around the "build portability" theme, so that you don't get issues as having the same code with the same buildscript working on one computer but not on another one (this is a known issue, we have VMs of Windows 98 machines since we couldn't get some of our Delphi applications compiling anywhere else). Because of this, it is also the best way to work on a project between people who use different IDEs since IDE-generated Ant scripts are hard to import into other IDEs, but all IDEs nowadays understand and support Maven (IntelliJ, Eclipse, and NetBeans). Even if you don't end up liking Maven, it ends up being the point of reference for all other modern builds tools.
Why you should use it
There are three things about Maven that are very nice.
Maven will (after you declare which ones you are using) download all the libraries that you use and the libraries that they use for you automatically. This is very nice, and makes dealing with lots of libraries ridiculously easy. This lets you avoid "dependency hell". It is similar to Apache Ant's Ivy.
It uses "Convention over Configuration" so that by default you don't need to define the tasks you want to do. You don't need to write a "compile", "test", "package", or "clean" step like you would have to do in Ant or a Makefile. Just put the files in the places in which Maven expects them and it should work off of the bat.
Maven also has lots of nice plug-ins that you can install that will handle many routine tasks from generating Java classes from an XSD schema using JAXB to measuring test coverage with Cobertura. Just add them to your pom.xml and they will integrate with everything else you want to do.
The initial learning curve is steep, but (nearly) every professional Java developer uses Maven or wishes they did. You should use Maven on every project although don't be surprised if it takes you a while to get used to it and that sometimes you wish you could just do things manually, since learning something new sometimes hurts. However, once you truly get used to Maven you will find that build management takes almost no time at all.
How to Start
The best place to start is "Maven in 5 Minutes". It will get you start with a project ready for you to code in with all the necessary files and folders set-up (yes, I recommend using the quickstart archetype, at least at first).
After you get started you'll want a better understanding over how the tool is intended to be used. For that "Better Builds with Maven" is the most thorough place to understand the guts of how it works, however, "Maven: The Complete Reference" is more up-to-date. Read the first one for understanding, but then use the second one for reference.