Anonymization refers to the practice of removing personally identifiable information from data, such that the individual people the data describes cannot be identified.
De-identification refers to the removal of obvious identifiers like names. This is generally insufficient to ensure anonymity, because inferences can be made based on the remaining data. This is called re-identification).
De-identification and anonymization are often conflated.
Anonymization is hard: Like cryptography, most people aren't qualified to do their own, but many established approaches exist (Deidentification versus anonymization)
"Linkage attacks" re-identify de-identified data. The classic example is Latanya Sweeney's identification of the personal health records of the Governor of Massachusetts.