Okay so lets get the stuff you'll need to run some Java programs!
Alright so first if your on Windows you'll need Java 2 Standard Edition SDK which can be downloaded here, just click on the one for JDK 6 Update 3. If your on a Mac you have it already.
When you are writing Java programs write them in a text editor that you can make the file a .java, the simplest ones like notepad are good.
You now need to set the PATH variable of your computer, follow this guide (windows) for that because I found it rather annoying. This is so you can use the Javac command in command prompt to run your program.
I have no clue how to change the PATH variable for Mac or if there even is a PATH variable but this may help because I believe that Mac runs on UNIX. Scroll down to Common Error Messages on UNIX Systems it has an explanation.
Now I find most of that annoying to the extreme but the books and tutorials I have read all say that you should start off running programs in the command line. Personally I think that you should download Eclipse. It is far easier to tell where you made errors and far easier to correct them resulting in faster learning I believe (if you can't find your mistakes and don't know what they are you can't correct them right?).
Here is a link to pretty much the ultimate reference guide for the Java API(explained later, bookmark this you'll need it eventually).
So now you have all that you will need.
Comment if you need help.