Most of the time police do not need a warrant in order to effectuate an arrest. Usually all they need is probable cause.
Probable cause can be hard to define. Different judges in different courts define it differently. The police must have enough justification or good reason to believe that you committed a crime in order to arrest you.
One court has defined probable cause as: when a reasonable man has the facts necessary to conclude that you probably committed a crime. If they have enough reason, or probable cause, they can arrest you.
There is one big exception to this rule. Even if they have probable cause to believe you committed a crime they cannot come to your house, go inside, and arrest you without a warrant.
In order to enter a person’s home the police must always have a warrant unless they receive consent or there is some other emergency situation that necessitates them going in. Otherwise, if they simply believe you committed a crime, they must get a warrant from the court in order to come and arrest you in your home.