If you have a juicer, juice the ginger.  I don't have a juicer, so I grated it with the Cuisinart.  You could use a blender, or just chop it into small chunks.  I didn't bother peeling it, but you do want to wash any loose dirt off the roots.

Combine all the ingredients EXCEPT the yeast in a big pot, and bring it to a boil.  It'll look like this:

As the water's heating, mix a teaspoon of sugar with 1/4 cup of warm tap water in a small bowl.  Sprinkle the yeast on top of the water, but don't stir it in.  It should take a while for the mixture to boil, but you want the yeast to get at least 15 minutes of activation time.

OK, your mixture has started boiling.  If you're using chunks/slices of ginger and lemon, you might want to let it boil for a few more (~5) minutes to extract more flavor.  Fill your sink with cold water - put ice cubes in it if you can - and carefully lower the hot pan into the cold water bath.  You want to cool it to lukewarm (below 100 degrees F) quickly, to avoid any bacterial contamination.

Once it's lukewarm, strain the chunks out of your liquid into your big jug/jar and stir in the activated yeast.  Seal LOOSELY, so that air can escape but bugs can't get in.  Easiest thing would probably be to put a clean cloth over the top.  You really, really do not want to seal this thing airtight, unless you fancy repainting your room with glass shards and ginger juice.

Watch the sides of the jar closely - after a few hours, you'll see some small bubbles.  Once you see a regular stream of bubbles (kind of like a carbonated soda), you know the yeast is happily working away.  If you never see any bubbles, then your yeast is probably dead or something went wrong - you should probably just toss it down the drain and try again.

Now it's time for bottling!  Go ahead and pour the brew into your plastic bottles, but not all the way to the top - leave some space at the top.  Seal the bottles tightly, and leave them in a warm spot (but not directly in the sun).

What's going on:  The yeast is busily eating up the sugar and expelling carbon dioxide as waste.  The CO2 has nowhere to go, so as pressure builds it gets "carbonated" into the liquid.  You're making it bubbly!  The warmer it is, the faster this process will go - at normal Berkeley temperatures you probably want to leave it overnight, but this weekend it was so hot that my batch was done in three hours.  You can test its progress by squeezing the plastic bottles - when they don't give at all, you know there's a lot of pressure built up inside.  Move them into the refrigerator to halt the process.  It's all done!

Serve cold, as you would any ginger beer.  It's very good mixed with iced tea.  There are also other, more adult options, if you feel like throwing more dietary restrictions out the window.

Sláinte!

A note on the alcohol content

Any time you ferment something, alcohol is a byproduct.  So yes, this does contain some alcohol - about 0.5% alcohol by volume, or 1 proof.  If that's unacceptable to you, there's a good alternative:  simply omit the step of adding the yeast and letting it sit.  You will still have a mighty strong and tasty drink.  You can either add bubbles with a carbonating device (ask Alex), or just mix the drink with some seltzer water over ice.