Summarize:

To tune a piano, you will have to take it (partially) apart to get to the strings. Examine your piano and determine what size screwdrivers or other tools you'll need to take it apart.  Expect the inside of the piano to be dusty. A feather duster or a few rags will probably come in handy. You also may want to grab a strong flashlight or other source of light. No matter how bright the room is where you're working, you'll have a hard time seeing once you get into the piano. Before you start working on the piano, play it a little with the panels off. Pay attention to which strings go with which keys, so you don't end up tuning the wrong string later on. It is relatively essential to have a decent understanding of music theory before you try to tune a piano. You should at least understand octaves and the relations of notes, because you aren't going to tune each note to perfect pitch. Rather, you tune the notes in relation to each other according to a concept called inharmonicity. The common tuning for most pianos is A440, meaning that the A4 is tuned so that sound waves vibrate at 440Hz. This is standard concert pitch in most of the Western world, although European tuning often is slightly higher at 442Hz.  A mid-treble note, such as middle A, typically has 3 strings per note. Mute the first 2 strings so that only the third can be heard. Tune that string to match the tone on your chromatic tuner, then tune the other 2 strings to match that string. The other strings are called the unisons. When tuning the unisons, do it by ear – don't tune the unisons using your software or chromatic tuner. Place your tuning lever on the top of the pin and make a very slight movement to turn the pin. You only want the tiniest of movements or you could snap the string, so you may want to practice first so you have good control of your tool.  Remember the phrase "righty tighty, lefty loosey." If you turn the pin right (clockwise), you raise the pitch. Turning it left (counterclockwise) will lower the pitch. Avoid wiggling or twisting the pin. If it becomes loose or damaged you'll need to call in a pro and have it repaired. Steady hands are important. Make a slight turn, then check the tone. Keep doing this until the sound of the string and the tone you are getting from your chromatic tuner or software program are in sync. Once you've found the right tone, you need to set the pin so it doesn't easily slip out of tune again. Tighten the pin just a hair with a slight turn clockwise, then turn it even more slightly counterclockwise to loosen it back to correct pitch. It takes practice to get this delicate movement correct. Don't expect the first piano you tune to stay that way. After you've tuned several pianos (or tuned the same piano several times), you'll have a better feel for how to set the pin correctly. Once you have middle A, use middle A to tune lower A. Then use lower A to tune the note one fifth above, and so on. Move along the keyboard using these intervals until the entire piano is tuned. Unless you are working with a severely neglected piano that hasn't been tuned in years, you'll typically find that you don't have to tune all the notes. Before you replace the panels, give the piano a quick play to make sure it sounds okay and all the notes have stayed in tune. Especially if this is your first time, you may find that you need to go back and do a little more work on it.Even for professional tuners, some pianos can take several days to tune correctly, especially if they haven't been tuned in quite some time or if they are played heavily. Patience is key if you want to tune a piano.
Gather basic tools to remove the piano's external panels. Get familiar with the strings and pins. Start with middle C. Turn the pin to tune the string. Set the pin. Tune in octaves relative to the first note tuned. Play the piano after you're done.