The term "matching donor" is colloquial for a person who has the same HLA type as the patient. HLA stands for Human Leukocyte Antigen. This is a group of genes which are essential for immunity to disease.
To determine the HLA type a cell sample is taken. In USA this is typically accomplished via a cheek swab where you take a Q-Tip and rub it on the inside of your cheek which deposits some cells on the Q-Tip which are examined in the lab to determine your HLA type. In Australia, Canada and India the cells are collected via a blood sample. The HLA types are mathematically coded and recorded. Matching is essentially comparing the codes between two HLA types.
Eligibility requirements vary from country to country. The requirements in USA are a little different from the requirements in Australia which are a little different than the requirements in UK.
Details of HLA typing are here.