API stands for 4′,6-diamidino-2-phenylindole and it is a fluorescent stain that binds to DNA in the nucleus. It is a stain used very often in fluorescence microscopy, when excited with ultraviolet light DAPI emits a blue light. This makes the stain very popular in fluorescence microscopy making the nuclei visible in a blue colour.
Scientists use DAPI staining for a variety of reasons. One common use is to visualize and analyze the nuclear structure and organization of cells. By staining the DNA with DAPI, scientists can observe the number, size, and shape of nuclei in a sample.
As an example in the picture below there are various cells which you can distinguish by the blue nuclei and the cytoskeleton stained with another dye that emits a green light. The picture was taken from thermofisher. It shows human iPSC staining using Mouse Anti Beta-Tubulin Monoclonal Antibody. The nuclei (blue) were labeled with Invitrogen NucBlue Fixed Cell ReadyProbes Reagent (Ready-to-use liquid DAPI formulation).
Red blood cells don’t have a nucleus and would therefore not be stained with DAPI.
The image below is taken from a PLOS one publication. All cells can be seen by the blue nuclei stained with DAPI showing in blue, the immune cells in red (CD45+ leukocytes) and the endocrine cells (Ki67+) in pink. The insulin is shown in green.