The fundamental construction step of building modified biological organisms is by using PCR.
The polymerase chain reaction (PCR) is used to amplify a single or few copies of a piece of DNA across several orders of magnitude, generating millions or more copies of the DNA piece. This is a chemical reaction involving a heat-stable DNA polymerase, such as Taq polymerase, which is an enzyme. Small volumes of solutions are used, usually under 100 microliters (uL). Using small amounts of solution saves on cost (the solutions are expensive) and can speed up the reaction.
An animation of PCR describes this process.
By using PCR, a customized piece of DNA can be replicated into an organism (typically a simple bacteria which reproduces rapidly). When the bacteria reproduces, it makes a copy with the modified DNA. After thousands of reproductions, a new colony of modified bacteria can be separated from the non-modified bacteria. The genetically modified organism is saved, studied, and/or kept as a friendly pet.