The rule
In 9-pin no-tap, a first ball at a full rack that knocks down 9 or more pins is a strike, worth 10 plus your next two balls, exactly like a normal strike. In 8-pin no-tap, 8 pins is enough.
What does not change
Only a first ball at a full rack counts. Your second ball still needs every remaining pin for a spare, so nine on your second ball is simply nine. In the tenth frame the rule applies to every ball thrown at a full rack, fill balls included.
A scoring example
You open with nine pins, then nine again, then 7 and 2. In regular bowling the first two frames would wait for a second ball. In 9-pin no-tap both are strikes: frame one is 10 + 10 + 7 = 27, frame two is 10 + 7 + 2 = 19, and frame three adds 9. After three frames you have 55.
The tenth frame in no-tap
A nine on the first ball of the tenth is a strike, so you get two more balls. Each of those that goes at a full rack counts as a strike with nine as well. Nine, nine and eight is X X 8 on the sheet: 28 pins in the tenth.
A full game: no-tap against regular scoring
First balls of 9, 9, 8, 10, 9, 7, 9, 8 and 9, a 1 after the 8 in frame three, the 7 and the 8 picked up as spares, and 9, 9, 8 in the tenth. In 9-pin no-tap that is 221: every nine on a first ball is a strike, seven of them in this game.
The same first balls under regular rules, with every single pin picked up for a spare and a 9 as the fill ball, make 177. The no-tap game is 44 pins higher, and that bowler did not miss a single spare.
Why leagues and tournaments play no-tap
Scores go up, strikes come more often and a frame with a nine-count ends after one ball, so games move faster. That makes no-tap a favourite for fun leagues, fundraisers and one-day tournaments, and it gives newer bowlers more strikes to celebrate.
No-tap and your average
No-tap scores run higher than regular scores and there is no official way to convert one into the other, so keep them apart from your regular average. In Pinfall you set no-tap per league, and those games stay out of your regular average and records.