ABV (Alcohol by Volume) is the percentage of a beverage's total volume that is pure ethanol — the number printed on virtually every beer, wine, and spirits label worldwide, and the standard way homebrewers report and compare a finished batch's strength.
Two formulas exist because one linear approximation isn't enough
ABV is calculated from the drop in gravity between OG and FG, but there isn't one single universally used formula — a simple linear formula ((OG - FG) x 131.25) is fast and accurate enough for normal-strength beer, while a more complex advanced formula accounts for the changing density of an alcohol-water-extract mixture and stays accurate at higher gravities where the simple formula drifts increasingly high. This site's ABV calculator reports both side by side rather than picking one.
ABW is a related but genuinely different number
ABV should never be confused with ABW (alcohol by weight) — the same beer reports a lower number in ABW terms than in ABV terms, since ethanol is lighter than water, and mixing the two up when comparing a beer's strength against a spirit or wine labeled in the other convention produces a genuinely misleading comparison, since ABW always reads lower than ABV for identical beer — mixing the two up makes a beer look weaker than it actually is.