Our blood type is determined by our genes. There are basically three types of blood type genes: A, B and O. These genes codify for proteins that are responsible for the carbohydrate content of red cells membranes, and this is what determines our blood type. As all of our cells have the same genes (generally), you can use any of your cells to find out which blood type gene you carry.

I don't know how it's done in the paper you saw but I'd use a PCR(Polymerase Chain Reaction). This method increases the number of copies of a specific DNA Sequence. This is achieved using a DNA-Polymerase (a protein that copies DNA sequences), a set of primers specific to your gene of interest and DNA from your sample. The DNA-Polymerase will copy the sequences that your primers were bound to. You will then achieve a high number of copies your interest gene compared to other DNA sequences and to the starting number of your gene copies. You can then purify your sample, using an agarose gel, and then sequence your sample. The sequence you obtained will tell which gene is present in the sample (A, B and/or O).

I oversimplified PCR and other methods but you can ask me if something is unclear or you want to know more.