American Wire Gauge (AWG) numbers run backwards: the smaller the number, the thicker the wire. Every three gauge numbers the cross-section roughly doubles, and every six the diameter roughly doubles, so 10 AWG has about twice the copper of 13 AWG and twice the diameter of 16 AWG. Sizes thicker than 1 AWG are written 1/0, 2/0, 3/0 and 4/0 ("one aught" to "four aught").
The three ampacity columns are the NEC's allowable currents for insulation rated 60, 75 and 90 °C, with no more than three current-carrying conductors together and a 30 °C (86 °F) surrounding temperature. Most home wiring is NM-B cable ("Romex"), which the NEC limits to the 60 °C column. The 75 °C column applies to wires such as THHN in conduit when the breaker and the equipment terminals are also rated 75 °C. The 90 °C column is mainly a starting point for adjustments, not a size you can pick a breaker from.
For the three smallest common sizes the breaker is capped below the table value. NEC 240.4(D) limits copper 14 AWG to a 15 A breaker, 12 AWG to 20 A and 10 AWG to 30 A, and aluminum 12 AWG to 15 A and 10 AWG to 25 A. Those cells are marked with an asterisk. In practice that gives the familiar home sizes: 14 AWG copper for 15 A circuits, 12 AWG for 20 A circuits and 10 AWG for 30 A circuits such as a dryer or water heater.
The resistance columns are for solid wire at 20 °C (68 °F); stranded wire of the same gauge is a little higher. Resistance is what causes voltage drop on long runs, which is why a circuit can need a bigger wire than the ampacity column alone suggests. For a specific circuit, use the wire size calculator, which checks both.
Use the gauge picker to jump to a size, switch between copper and aluminum, or type a current to see the smallest wire for it. The print button prints the chart on its own, sized to fit one page.