Basically, they're straightforward to any other painting program such as MacPaint, Canvas, etc. If you have no knowledge of these types of programs, then where have you been? It's not too tough.. go figure it out. There's no need to go into the explanation of all these tools.
Notes:
*The only thing to note is that the eraser tool will paint a swath of the current selected background color (in your palette). So, if you wish to erase something, you might have to first select a different color, such as White.
†The pencil tool is not as sensitive to movement as you might normally be familiar with in other painting programs. To get the accuracy you want, slow your drawing speed down. The reason for this is to decrease bandwidth consumption from HomerPaint sending coordinates for every pixel you draw.
The Little Floater
That little floater window is where HomerPaint lists the user who is currently painting on the channel which your HomerPaint is painting to.
  <---- Floater in question...
Color Palette
The color palette shows you a list of possible colors you can select from. It also shows you the current Foreground, Background and Fill colors. To select a color, you simply point your arrow to the little color you'd like and click on it.. bingo!
  The small box on the left side at the bottom of the colors (where it
is currently black) is your current Foreground Color.
The small box on the right side at the bottom of the colors (where it
is currently red-orange) is your current Background Color.
The small rectangle on the bottom of the palette (where it is currently
pink) is your current Fill Color.
< Obviously if you're not using a color monitor, this isn't gonna be too important to you, but it's always fun to read documentation though isn't it? Yeah.. right, find a hobby.. >
In any case, it sucks to go and select foreground/background/fill colors with the color picker. Matter of fact, it also sucks picking your nose, nasty habit. So.. what you can do is use the handy little shortcuts on selecting colors.
Color Selecting Shortcuts
Foreground Color:
The Foreground color is selected by clicking on any color without holding any special keys. It defaults to changing the Foreground color with selections.
Background Color:
The Background color can be easily changed while holding down your Command key while selecting a new color from the palette with your mouse. (This color also applies to the Eraser tool and to the Selection tool when a selection is moved).
Fill Color:
The Fill Color can be easily changed by holding down your Shift key while selecting a new color from the palette with your mouse.