The definition varies slightly from law to law.
The EU Data Protection Directive defines, in Article 2:
(a) 'personal data' shall mean any information relating to an identified or identifiable natural person ('data subject'); an identifiable person is one who can be identified, directly or indirectly, in particular by reference to an identification number or to one or more factors specific to his physical, physiological, mental, economic, cultural or social identity;
It also includes in Recital 26 the following:
(26) Whereas the principles of protection must apply to any information concerning an identified or identifiable person; whereas, to determine whether a person is identifiable, account should be taken of all the means likely reasonably to be used either by the controller or by any other person to identify the said person; whereas the principles of protection shall not apply to data rendered anonymous in such a way that the data subject is no longer identifiable; whereas codes of conduct within the meaning of Article 27 may be a useful instrument for providing guidance as to the ways in which data may be rendered anonymous and retained in a form in which identification of the data subject is no longer possible;
This definition is then implemented, in various ways, in the Member States laws. Of course, this definition has to be interpreted in light of EU jurisprudence on the matter.
In the Swiss Federal Act on Data Protection, it says (unofficial English translation, Article 3):
personal data (data): all information relating to an identified or identifiable person;
(note the implicit leap from natural person to legal person, which is cemented later in that Act)
The EU-US Privacy Shield Framework defines (Art 8, page 3):
“Personal data” and “personal information” are data about an identified or identifiable individual that are within the scope of the Directive, received by an organization in the United States from the European Union, and recorded in any form.
The CH-US Privacy Shield Framework defines (Art 8, page 15):
“Personal data” and “personal information” are data about an identified or identifiable individual that are within the scope of the F[ederal] A[ct on] D[ata] P[rotection], received by an organization in the United States from Switzerland, and recorded in any form.
So in both cases, the Privacy Shield Frameworks lob back the question to the Directive or the Swiss act.