Step 1: Acquire ingredients
True balsamic vinegar tradizionale is made from cooked grape juice slowly aged in a variety of wooden casks for a minimum of 12 years, developing an intense and complex set of flavors and slowly evaporating excess water. The longer the aging the better the resulting vinegar.
The balsamic you and I can afford is actually sweetened wine vinegar combined with caramel color. A good version of the inexpensive stuff is sweetened with reduced grape must (whole juice), and will have 3 ingredients on the label. A bad version will have a longer label list, probably including nasties like high fructose corn syrup. Read the label carefully. We used Trader Joe's balsamic, which is a reasonable balsamic of the three-ingredient variety.
So, you will need:
1 bottle balsamic vinegar
1 heavy-bottomed pot
1 stirring implement
some free time
Step 2: Cook over low heat
You don't want to simmer the vinegar, so if you see bubbles turn it down. What you really want to do is VERY SLOWLY evaporate excess water to concentrate the flavors, so expect that this will go on for an hour or two depending on your volume.
Check and stir periodically. Keep an eye on the level in the pot for an idea of how far your liquid has reduced. As the thermal mass decreases with volume, you'll likely need to lower the heat even further as the vinegar reduces.
Vent your kitchen to avoid filling it with lingering vinegar fumes.
Step 3: Decant, save, and use
We started with 1L of vinegar and ended up with 250mL for a 4x concentration. This turns out to be a pretty good concentration for drizzling over food, but is too syrupy to be mistaken for straight vinegar. A 2x concentration would probably work better if you prefer to maintain a less syrupy texture.
Immediately after cooking there we found a slight bitterness to the vinegar reduction, but this disappeared after a couple of days storage. I'd recommend letting the flavors mingle for a week before using your reduction.
So far we've drizzled the 4x reduction over apples, omelettes, salads, tomatoes, roasted vegetables, and meats; it's multi-purpose. Try it on virtually anything.