If you apply a force of $1N$ of a body of $1kg$, it will move with $1\frac{m}{s^2}$ acceleration. Alternatively, if a body of 1kg is moving with acceleration of $1\frac{m}{s^2}$ then the force acting on the body is $1N$.
But if a body has a constant speed then there is not net force acting on it.
A force can bring acceleration to a body motion, and acceleration means change in velocity.
According to newtons second law of motion,
$F =\frac{dP}{dt}$
$F=m\frac{dv}{dt}+v\frac{dm}{dt}$
Considering, $\frac{dm}{dt}=0$, then
$F=ma$
When you punch someone, your fist has some certain momentum at the time of impact. When you fist lands on your opponent, according to Newton's third law of motion, your fist apply force on the opponent and the opponent apply equal force back at your fist.
Its all about the impulse, $I=F\Delta{T}=\Delta{P}$
If you increase the time of impact, then the force will decrease. For a short amount of time, impulse can be enormously large in magnitude.
That is also why the fielders ,in cricket, swing their hands in the direction of the ball's trajectory so that they could decrease the impulse. If a fielder doesn't move his hands and tries to catch the ball with still hands, he might get hurt pretty bad, as in this case the impulse in high due to short time interval of impact.