Police officers don't need a warrant to make a search "incident to arrest." After an arrest, officers have the right to protect themselves by searching for weapons and to protect the legal case against the suspect by searching for evidence that the suspect might try to destroy. Assuming that the officer has probable cause to make the arrest in the first place, a search of the person and the person's surroundings following the arrest is generally valid, and any evidence uncovered is typically admissible at trial.
If police officers have probable cause, they can typically arrest and search suspects who've committed minor offenses that usually result in citations rather than arrests. (Virginia v. Moore, U.S. Sup. Ct. 2009.)
Example: Police Officer Leigh issues Miller a traffic ticket for driving with a broken tail light. After noticing that Miller's driver's license is suspended, Leigh arrests Miller, even though the law authorized him only to cite and release her. Leigh searches Miller incident to the arrest and finds illegal drugs in her purse. He refers the case to the prosecution with an additional charge of possession of illegal drugs. Leigh probably had the right to arrest and search Miller. Because Leigh had probable cause to make an arrest, the search is likely constitutional even though Leigh could have issued a citation to Miller rather than arrest her.
An officer cannot search your dwelling or car when you're arrested elsewhere, such as on the street or at the mall. To justify a search as incident to an arrest, a spatial relationship must exist between the arrest and the search. The general rule is that the police may search the arrested person and the area within that person's immediate control. For example, an arresting officer can often search not only a suspect's clothes, but also his or her wallet or purse. (Chimel v. California, U.S. Sup. Ct. 1969; see Cellphone Searches After Arrest.)
Example: Officer Montoya arrests Sarah for driving under the influence of illegal drugs. Before taking Sarah to jail, Montoya takes Sarah's key and enters her apartment. Inside, Officer Montoya finds a number of computers. He checks their serial numbers and finds out that they have been stolen. Officer Montoya seizes the computers as evidence and adds possession of stolen property to the charges against Sarah. The computers are not admissible in evidence. The officer didn't have a warrant allowing him to enter Sarah's apartment. Because Officer Montoya had no right to be inside the house in the first place, it doesn't matter that the computers were in plain view once he was inside.
When officers arrest someone outside the home and have no basis for making a protective sweep, they may offer to let the suspect go inside to get a change of clothes or feed a pet before heading to jail. This offer may be an attempt to expand the scope of a search. While accompanying the suspect inside the residence, officers can seize whatever may be in plain view (for instance, drugs). Thus, many suspects refuse this kind of invitation, and instead rely on their friends if they need clothes or pet care.