What can you do with Java?
Pretty much everything, in fact! Java is a solution if all other methods fail.
How does it work?
Java is an object-oriented programming language in the syntax-style of C/C++, invented by Sun Microsystems. The idea behind is, it should run on every system that implements the so-called Java Virtual Machine (JVM), a safe environment in which the compiled Java classes are running.
The Java-Applet is a special kind of Java environment. It is a box displayed in the browser, embedded into an HTML page.
In fact, except for some of the syntax, and the name of course, there is no connection between these two languages. They serve a different purpose and are technically miles apart.
"Write once, debug everywhere!" is not quite true, but there are various implementations of the JVM, depending on which browser is used. Some applets might run fine on the Internet Explorer, and crash inside Netscape.
How fast does it run in browsers?
Since the Java Virtual Machine is mostly not initialized on browser-startup, loading an applet always takes some seconds, which can be annoying to the user if there's not much of a need for the applet. Of course, there's also the classed that have to be downloaded by the user, and these might all together not be of a small size.
Is Java ultra-fast when a big display area is used?
Since Java is executing on the client, this largely depends on the client hardware, but in general: no, right now you can't do any super-fast games in full-screen display with Java. Java is not compiled to machine code, it's compiled to a way that it still has to be interpreted by the JVM.