The nurse is educating a client who is about to undergo a stem cell transplant mentioning that a single stem cells can give rise to many cells needed for normal tissue repair. with stem cell division, one daughter cell retains the stem cell characteristics while the other daughter cell becomes which type of cell leading to terminal differentiation?

The answer is: progenitor cell.
When a stem cell divides, it originates another stem cell and a progenitor cell.
A progenitor cell is similar to a stem cell but has the tendency to differentiate into a specific type of cell (it already has a 'target' function- for example: skin, blood, muscle cells...). It's more specific than a stem cell and is closer to differentiate. Something very important that can be told to the patient is that this cells can't replicate indefinitely like stem cells, they have a limit number of divisions.

Being a cell having totipotency one cell can become any kind of cell. The other cell will become terminally differentiated. A Terminally Differentiated cell or TD cell will become cell having particular function. It will lost its capacity irreversibly to proliferate.