If there is an Order in place, usually a Child Arrangements Order (formerly a Residence Order) then a child can be taken abroad for up to a month without needing the written consent of the other parent (Section 13 (2) of the Children Act 1989).
If there isn't a court order in place, then do you have what is termed parental responsibility? If both parents share parental responsibility, then you still need to get written consent from the other parent in order to take your child out of the United Kingdom. (Section 13 (1) of the Children Act 1989). If you don't, it could lead to you committing an offence of abduction for which you can be fined, imprisoned or both. But if consent is unreasonably withheld, then an application to Court can be made.
The Judge will take into account the individual circumstances of each family, and if permission is given (which it often is) then you need to provide your specific travel details, including your travel dates, the address details for where the child will be staying and any flight numbers. The Court rarely denies permission to take a child on holiday abroad where there is an existing relationship between the parent and child and the plans are reasonable in all of the circumstances. If the Court does deny permission, it's usually in circumstances where the plans are patently not in the child’s best interests or where the Court deems the child may not be returned to the country.
If only the mother has parental responsibility, and again there are no Court orders in place, then permission is not necessarily needed to take a child abroad on holiday. That being said, and with your child’s best interests at heart, consultation should always take place with the other parent (if they are in regular contact with the child) in order to reach an agreement that's right and works for everyone. As a father without parental responsibility, if you don't agree to your child being taken abroad, you can apply to the Court for a Parental Responsibility Order and a Prohibited Steps Order to prevent the trip.
If it's the case that another member of the family, such as a grandparent, wants to take a child abroad, then it is worth noting that permission will be needed from both parents who have parental responsibility, not just from one.