Find your breaker in the first table and read across to the column for your wire. With NM-B cable ("Romex"), the usual house wiring, use a 60 °C column: NEC 334.80(A) limits NM-B to its 60 °C ampacity, and the same goes for UF cable and for any breaker or device whose terminals are rated 60 °C. The 75 °C columns are for individual wires such as THHN or XHHW in conduit, and only when the terminals at both ends, the breaker and the equipment, are rated 75 °C. If that can't be confirmed, the 60 °C column applies.
A few rows show how it works. A 30 A breaker takes 10 AWG copper or 8 AWG aluminum in every column. At 70 A, copper needs 4 AWG in both the 60 and 75 °C columns, while aluminum needs 2 AWG at 60 °C or 3 AWG at 75 °C. At 100 A it is 1 AWG copper at 60 °C or 3 AWG at 75 °C. At 200 A, 3/0 copper at 75 °C is enough, while nothing up to 4/0 works at 60 °C or in aluminum.
Table 310.16 rates 12 AWG copper at 25 A in the 75 °C column, yet a 25 A breaker needs 10 AWG. NEC 240.4(D) sets a lower breaker limit for the smallest sizes, and in the 75 °C columns that pushes the 20 to 35 A rows up a size; without it, 10 AWG copper (35 A at 75 °C) would also cover a 35 A breaker. That's why the 20 to 35 A rows name the same size in the 60 and 75 °C columns for each metal. The ampacity table marks each capped value with an asterisk.
Every row follows one rule: the breaker protects the wire. A breaker only trips when the current goes over its rating, so the wire's allowed current must be at least the breaker rating, or the wire could be overloaded with the breaker never tripping. That's why you choose the wire from the breaker.
NEC 240.4(B) sometimes allows the next standard breaker size above a wire's ampacity, when the ampacity doesn't match a standard rating and the circuit doesn't supply more than one receptacle for plug-in portable loads. This chart doesn't use that allowance: every wire it lists is rated at least the full breaker size. Whether 240.4(B) fits a particular circuit is for your electrician or inspector to decide.
The ampacity-by-size table is the same Table 310.16 data as the wire gauge chart on the home page, with all six columns side by side. Pick a breaker from the 60 and 75 °C columns only. The 90 °C values come into play when a wire is derated for crowding or heat, as the figure the reductions start from.
What the chart doesn't cover: length, crowding and heat. It sizes wire for heat, not voltage drop, so a long run can need a bigger size. On a 60 A circuit at 240 V with 60 °C wire, 4 AWG copper covers a run of up to 150 feet within 3%; at 200 feet it would drop 3.02%, so the size goes to 3 AWG. For your own distance, use the wire size calculator. More than three current-carrying conductors together, or surroundings above 30 °C (86 °F), call for the NEC 310.15 adjustments.
A home's main service, or a feeder that carries the whole home's load, is a separate case: NEC 310.12 lets those conductors be sized at 83% of the rating. This chart doesn't apply that rule; the 100 amp wire size page explains when it applies and what it allows.