What is Newton's third law?
For every action there is an equal and opposite reaction. More precisely, if body A exerts a force on body B, then B exerts a force on A of the same size along the same line but in the opposite direction. The two forces always form a pair, always act on two different objects, and always appear and disappear together. There is no delay and no exception at everyday scales.
If the forces are equal, why does one object move more?
Because equal forces produce very different accelerations on different masses, which is Newton's second law doing the work. Push a supermarket trolley and you push the trolley exactly as hard as it pushes you, but you weigh far more than it does, so it accelerates away while you barely move. Fire a rifle and the bullet and the rifle receive identical impulses; the bullet is a thousand times lighter, so it leaves at high speed while the rifle merely thumps your shoulder.
Why don't the action and reaction forces cancel out?
Because they act on different objects. Forces only cancel when they act on the same body. A book resting on a table pushes down on the table and the table pushes up on the book — those are a third-law pair acting on two different things. What holds the book still is a separate balance: the table's upward push on the book cancels the Earth's downward pull on the book. Mixing these two ideas up is the single most common error in the topic.
How does a rocket work in a vacuum with nothing to push against?
It pushes against its own exhaust. The engine throws mass backwards at high speed, and the third law gives an equal forward force on the rocket. Nothing external is needed, which is why rockets work in space and jet engines do not. This is also momentum conservation viewed from a different angle: the total momentum of rocket plus exhaust stays constant, so as the exhaust gains momentum one way the rocket gains it the other.
How do you walk, according to Newton's third law?
You push backwards on the ground with your foot, and the ground pushes forwards on you with an equal force. That forward push from the ground is what accelerates you. The reason ice is difficult is not that the third law stops applying but that friction is too low to let you push backwards hard enough — the pair still exists, it is just very small. Swimming, rowing and driving all work the same way.