A Legal Admission
Generally, a legal admission to the United States is after a person, a non-United States Citizen presents themselves at a port of entry seeking "admission" to enter the United States. A Port of Entry is a place designated by the Attorney General and is run by the Customs and Border Protection.
Inspection and Admission
After a person presents themselves to an immigration officer at a port of entry, the person is inspected. This means that the immigration officer looks at their travel documents, passport, identity documents, visa, and determines if the person is admissible to the United States. The law states that if you are not a United States Citizen, you are inadmissible to the US unless you fit one of the many exceptions. The officer determines those exceptions and makes sure that the non-USC can enter the United States.