'Pro-' is a general Latin preposition and English prefix with many meanings, but often in biology, as in this case, it means "before". A proacinar cell is not yet an acinar cell (a cell of the acinus) but will be. A related usage occurs with peptides such as proinsulin.
In Latin, "acinus" means a cluster of grapes. You can find 'acini' in many places, notably exocrine glands, but in this case the pancreas is discussed. With the help of Ms. Elbakyan, they tell quite a beautiful story, but the answer to this question is mostly in the first sentence of the abstract: the stem cells (multipotent pancreatic progenitor cells) can become pro-acinar cells (which become secretory tissue of the acini, producing the pancreatic digestive enzymes), or else are reserved as bipotent progenitors that create the ducts and the islets of Langerhans.