Police can enter your home without your permission but only with a warrant or if they witness someone being harmed or about to be killed etc.
If police show up at your house and there's nobody in danger then the police have to have your permission to enter your house or they have to have a warrant to enter your house.
Even if you or someone that lives in the house is there and they have a warrant for the person they still cannot enter the house to simply make the arrest on the warrant.
To enter the house without it being an emergency and without your permission then the police have to get a warrant signed by the judge.
With the legal warrant the police can then enter your house and can even break down the door to enter the house without you or anyone else's permission.
If a police officer arrives at your house on a call and someone is inside about to kill someone or harm someone then they can legally enter the house under those circumstances to stop the threat.
For example if someone in your house is threatening to harm you or someone takes you into their house and is about to harm you and the police arrive.
The police can enter the house to save the person who is being threatened or harmed or about to be killed.