A patient identifier is any unique number that can identify a patient. Examples are a Medical Record Number, a National ID, a Social Security Number, a driver's license number, etc. A patient can have any number of identifiers. The Patient Identifier Type table defines what type of identifiers are collected in your system.
A patient can only have multiple identifiers of each type defined in your system. E.g. a patient could have 5 identifiers of type of "Medical Record Number" because they were seen at 5 different hospitals that collected 5 different types of ids.
The patient search screen searches across all identifier that are still active for a patient.
New identifier types are generally created if they have different characteristics. E.g. one identifier can be only a string of numbers, another is a number with a hyphen plus a checkdigit, etc.
Also see Adding A New Patient Identifier Type