While the term "duplication" is adequate, we typically prefer the term "replicate" when referring to the process of making exact copies of DNA. This is because the word replicate has retained most of its "scientific" definition, whereas duplicate has fallen into common usage and can be misconstrued as meaning "copy or reproduction," without the emphasis on that reproduction being 100% identical to the original.
DNA replication typically occurs as a part of the cell's division process, whether that process is mitosis or meiosis depending on the cell. Replication takes place inside the nucleus of a eukaryotic cell. There can be some nuanced differences in the exact conditions of replication depending on the type of cell, but the general procedure is for a helicase enzyme to split the hydrogen bonds between the two rungs of the double helix, and then a series of other enyzmes coordinate to replicate the strands while managing side effects such as the strain on the remaining double helix and the reassembly of the single strands into double strands.
Replication takes place during the interphase portion of a cell's life cycle, and could be considered the first step in the process of eventual cell division.