There are three ways I can break the law:
1. actively do something myself that is against the law: I murder my wife
2. (More common in America) - I pay somebody to do it for me. I am miles away at the time, making sure I have a good alibi!
In (1), I commit the crime myself. In (2), I conspire with someone, 'authorize' them to commit a crime.
The law combines these and calls them "acts of commission".
The law also states that in certain situations, a person has a legal duty to act. So - if a child is drowning in a lake, and I, a stranger, take no action to help save the child, I am not breaking the law. (The newspapers would make mincemeat of me though!).
However, if I were the child's uncle, then as a blood relative, I do have a legal obligation, a duty of care, to go to the assistance of the child. Failure to try to rescue the child -failure to act - is an Act of Omission.
I found this example on the Internet:
In R v Pittwood (1902), the defendant was convicted of gross negligence manslaughter after he failed to close the gate on a level crossing as he was contracted to do. This caused a train to collide with a hay cart, and the court ruled that "a man might incur criminal liability from failure to comply with a duty arising out of contract."
I am guilty of an act of commission if I hide an escaped convict in my house.
If I know that someone else is hiding an escaped convict, and when I am questioned by the police and asked if I know where the escaped convict is, and I don't tell them, then I am guilty of two Acts of Omission: (1) Failure to help police with their enquiries and (2) withholding information.