First off, make a "plant" for your ginger beer. Stir the yeast into the warm water until dissolved. Add in 1 tablespoon freshly grated ginger, 1 tablespoon sugar, the lemon juice, sliced jalapeno and stir to combine. The jalapeno will give your ginger beer that kick you can feel in the back of your throat -- if you don't roll like that, omit it. Pour into a glass jar -- one that's large enough for the liquid to fit comfortably, with a bit of extra space. Cover with a clean, dry kitchen towel and secure it over the jar with a rubber band. Place jar in the warmest place in your house. Next to your heater, near the refrigerator, or by a heat vent.
Every day for the next week you'll have to "feed" your ginger beer. First off, feel the bottle -- it should be slightly warm. If it's too cold your yeast will go into hibernation, and if it's too hot it could kill your yeast. Take off the towel and add another tablespoon of grated ginger, and another tablespoon of sugar. Stir until sugar dissolves, then replace the towel and put your plant back in a warm place. Do this every day for a week -- think of it as your neighbor's dog you've promised to dog sit.
After about a week, you should see small bubbles floating to the surface of your plant. You can certainly keep your plant at this stage longer; the more you feed it, the more concentrated the ginger flavor will become. You can adjust flavors later!
Now it's time to bottle. Think ahead to how many bottles of ginger beer you'll want to make. Make sure to use PLASTIC soda bottles -- glass bottles could explode from carbonation, which would not be pretty. Estimate how much water you'll need to fill these bottles 3/4 of the way full, then boil it to purify. Dissolve enough sugar into the water that it tastes very sweet -- as sweet as soda. You can adjust this later as well.
Using a cheesecloth, strain the plant out into a large measuring cup, or a bowl. Using a funnel, add about a cup of the plant liquid to each clean, dry soda bottle -- more if you want it stronger, less if you want it less intense. Add sweet water to the bottles until they are 3/4 of the way full, then stir with a chopstick to combine. you can dip your finger in and taste here to see if the mixture needs more ginger. If so, add more plant liquid. Don't worry if it seems too sweet: the yeast will eat the sugar and turn it to alcohol, so most of it will disappear. You can add it back later.
Seal the bottles tightly with their caps and place them back in the warm place you had your plant. Squeeze the bottles once a day to test how they're carbonating. After a few days they should be hard to compress; when they are impossible to compress at all, slowly start to unscrew the cap just until the carbonation begins to release -- do not open it all the way! Do this whenever you can't compress the bottle at all.
After a week and a half to two weeks, the yeast should have eaten up most of the sugar in the bottle. This means your ginger beer is ready to open up and taste! If you have multiple bottles, open one up and taste test. Add more sugar or lemon juice if you think your ginger beer needs it. Serve ice cold with citrus, and a rum float if you're feeling dangerous. Make sure to consume the whole bottle within 24 hours once you've opened it -- feel free it enlist a friend here. It's impossible to gauge the alcohol content of your ginger beer, but it should be a bit less than a light beer. Enjoy!