Parental responsibility means according to the Children Act "all the rights, duties, powers, responsibilities and authority which by law a parent of a child has in relation to the child and his property".
If you are the child's mother then you automatically acquire parental responsibility when your child is born. If you are the child's father and were married to the mother at the time of the child's birth, or if after 1st December 2003 your name is on the birth certificate then you will automatically have parental responsibility.
If you are the child's father you may acquire parental responsibility by agreement with the mother - or by an application to the court. The court considers various factors in making a parental responsibility order and a family law solicitor will advise you on this application. Other people - including step parents - are entitled to acquire parental responsibility provided that both parents agree and the court approves.