Anonymity is derived from the Greek word ανωνυμία, meaning "without a name" or "namelessness". In colloquial use, anonymous typically refers to a person, and often means that the personal identity, or personally identifiable information of that person is not known.
More strictly, and in reference to an arbitrary element (e.g. a human, an object, a computer), within a well-defined set (called the "anonymity set"), "anonymity" of that element refers to the property of that element of not being identifiable within this set. If it is not identifiable, then the element is said to be "anonymous".
The term "anonymous message" typically refers to message (which is, for example, transmitted over some form of a network) that does not carry any information about its sender and its intended recipient. It is therefore unclear if multiple such messages have been sent by the same sender or if they have the same intended recipient.
Sometimes it is desired that a person can establish a long-term relationship (such as a reputation) with some other entity, without his/her personal identity being disclosed to that entity. In this case, it may be useful for the person to establish a unique identifier, called a pseudonym, with the other entity. Examples of pseudonyms are nicknames, credit card numbers, student numbers, bank account numbers, and IP addresses. A pseudonym enables the other entity to link different messages from the same person and, thereby, the maintenance of a long-term relationship. Although typically pseudonyms do not contain personally identifying information, communication that is based on pseudonyms is often not classified as "anonymous", but as "pseudonymous" instead. Indeed, in some contexts, anonymity and pseudonymity are separate concepts.
However, in other contexts what matters is that both anonymity and pseudonymity are concepts that are, among other things, concerned with hiding a person's legal identity. In such contexts people may not distinguish between anonymity and pseudonymity.
The problem of determining whether or not the identity of a communication partner is the same as one previously encountered is the problem of authentication.