Before you begin, you should make sure that you have everything on the list of necessary supplies.
I use StarSan, which must be mixed with water prior to use (1oz StarSan to 5 gallons of water). Mix the sanitizer with water inside your bucket, and soak the supplies that you will be using for 15 seconds. The StarSan solution is very foamy, but do not fear the foam! It is perfectly safe for contact with your brew.
Start with the cider using the funnel, then sprinkle in the appropriate amount of nutrient and yeast on top. Typically a packet of champagne yeast will contain enough for 5 gallons, so you can use 1/5th of the packet for your gallon jug. Add the nutrient accoding to instructions on the package.
Make sure that you use sanitized water to prevent any unwanted wild yeast or bacteria from entering your brew. The airlock will allow carbon dioxide to leave the jug during the fermentation process while preventing anything from getting in.
The yeast will eat the yeast nutrient and convert the sugar to alcohol and carbon dioxide.
Don't foget to sanitize everything beforehand, including the bottles! For clearer cider, take care to siphon from the top of the jug.
These will add carbonation. Just pop a few tablets into each bottle (more tablets = more bubbles; you'll typically use 3-5 per bottle). After capping, store in a cool dark place for another 2 weeks to allow the tablets to dissolve and bubbles to form.
Use your bottle-capping tool to clamp sanitized bottle caps to the tops of your bottles. Put some elbow grease into it!
If you added conditioning tablets to your cider in step 6, this will happen two weeks after bottling day. If not, enjoy your delicious cider!