0] Java in a nutshell. (http://oreilly.com/)
1] Let's see.  There are a few things.  One is the Object Oriented nature of Java.  That isn't that hard to get used to, though.  Another thing is how everything is structured.  It's not really complicated and explained in said book.
2] Not going to explain classes as it's in that book.  As for the others:
* Private refers to a method or class that cannot be really accessed.  For example, a private method cannot be called outside that class.  Mostly used for internal class stuff.
* Public refers to a method that can be called inside the class, inside the package and outside the package.
* Protected refers to stuff that can only be called inside the package, but not outside.
* Static is slightly complicated, as it requires knowledge of something you don't already know, so here we go.  Say I have a class called MyClass.  Say all MyClass did was store a number.  Not really class worthy as there are "native" things like int to take care of that (Although there is the Integer class, but that's beside the point right now), but nevertheless, let's try it.  So, the "constructer" of MyClass takes one argument.  It can have more than one constructer, btw.  So anyway, to create an "instance" of the class, I do this:
MyClass variableName = new MyClass(5);
Which creates a variable called variableName which is an instance of the class MyClass.  Now then, any method not declared static applies to the object itself.  So, the static method variableName.toString() would really be MyClass.toString().  In other words, static references the class, whereas no static references the object.

Final words: Just read the book. ;-)

I recently joined a robotics club at school, and joined the programming team.  However, as it turns out, the robot will NOT be programmed with either Axe or TI-Basic (blasphemy!) and will be programmed in Java instead.

Yup. Java is popular so people are using it where it does not belong.

What are some common 'gotchas' that might trip me up, especially given that I'm used to coding calculators?

Surprisingly enough the biggest gotcha is not the Java - it's the culture around Java. It's decidely unhealthy.

All books and courses will try to teach you "Java way": where extensibility and testability is paramount and efficiency is secondary concern. This works fine when you develop some complex and slow system (where the ability to create something working is often more important then efficiency of the result: if the thing does not work at all then it does not matter if it's efficient or not), but for simple things "Java way" often produces horrible results.

If the task is simple and does not need extensibility then often it's better to just use array or matrix (as if you worked with Basic) rather then hierarchy of objects (as "true Java way" teaches).

Why is that? The problem here is dynamic memory and garbage collection: the more objects you create the more unpredictable the timing becomes - and this is NOT a good thing for robotics. Books will say that lots of temporary objects is not a problem: GC is sophisticated enough and will take care of everything... and this is just not true.

It does not matter how big and elaborate your collection of objects becomes as long as it's essentially static (so you don't allocate anything on constant basis and so garbage collector is not forced to clean up everything constantly), but when you start creating myriads of new objects in response to exteranl stimulus - you are in trouble. I've seen the case where simple logging spent more resources then actual work in automation system because it used standard library - and something like printf("Parameter f is %3.0lg",f); generates dozens of objects (starting from the call itself: it actually creates anonymous array). Jerky reaction become smooth and fast when said logging was replaced with home-brewn procedures which used pre-allocated arrays.

I guess maybe they absolutely want the team to code it in Java. Is it for some sort of java classroom and the teacher decided that it would be in java? That said, in the end, if what is produced is great, I think that's what counts the most, not how it was made. Back in 2004 I remember cases where TI-83+ BASIC games were put down by elitist ASM fanboys just because they were written in BASIC, even if the ASM counterparts weren't even as great (except for speed).

True.  In this case, it's basically ease of coding vs. speed requirements.  I program in Java all the time, but usually won't release something in it unless it can afford to be slow, or doesn't do labor intensive things.

Java can be programmed for mostly anything, as of recently. even platform-specific electronics can sometimes run java. if there's one thing java's good at, it's portability its ability for old versions of programs to still work today. oracle is so paranoid about java backwards-compatibility, that some of the constants aren't all uppercase (Color.Green == Color.GREEN). phones can run java, computers can obviously run java, i've even see a VC recorder run java. but what calcforth said is true. it really isn't a good language. and if you try to do something out-of-normalcy in the language, you'll get hit for it because java is all about standards.

Are there exceptions, though? I remember a year ago the Java applet for IRC on Omni website stopped working completly for many people. After a few java updates, it seemed to work fine, though.

Java doesn't work on everything.  It must have a JVM or Java Virtual Machine built inside to run the Java ByteCode.

Java can be programmed for mostly anything, as of recently.

You have very peculiar definition of recently. Java Card was introduced 15 years ago.

if there's one thing java's good at, it's portability its ability for old versions of programs to still work today.

Not true as well. C beats Java by wide margin: lots of programs from 1970th can be compiled and run today. And if we are talking about unmodified code, then the answer is two words: Java ME.

if you try to do something out-of-normalcy in the language, you'll get hit for it because java is all about standards.

Yup. Really, the only advantage of Java is the fact that muddleheads can write half-working code. If you give some specs and Java IDE to bunch of monkeys then there are some non-zero probability that they'll produce some kind of more-or-less working program. With C++, Haskell or even Pything this probablity is very close to zero so you are constantly faced with the dilema: what to do with people who are ruining the projects but can not be fired because of labour legislation?

I recently joined a robotics club at school, and joined the programming team.  However, as it turns out, the robot will NOT be programmed with either Axe or TI-Basic (blasphemy!) and will be programmed in Java instead.

Is there any chance you can use a different language, or is this for something like FIRST where you're given language requirements? If you can switch languages, I'd recommend something like Python (if possible). It's easy to write and is extremely powerful. If all else fails, you can switch to mechanics. Then you can blame the programmers for everything that goes wrong