Can someone please explain in terms that a simple Jew can understand, the definition of the kabbalistic concept called a Partzuf?
In order properly answer the question it would help to know in what context you see the word, so that the explanation can fit the context. However, in general, the Alter Rebbe of Lubavitch explains it like this:
The Sefiros spread out into specific detail expression 9 times 9, each level having a head, middle and end, plus five Chassadim - levels of kindness - that create the expansion become 248.
In other words, the Sefiros combine with each other to create a more complex structure. He gives a parable for this from how the soul is enclothed in the body. There is an ability to see, an ability to hear, and ability to walk, etc.
This combination creates the ability for there to be an advantage and connection in the end to the beginning. In other words, the head has intellectual abilities that are certainly higher than the ability to walk. But still, the head cannot walk anywhere, and thus lacks an ability of the feet.
Similarly Partzufim create a reality and connection with Hashem that isn't possible in the strict orderliness of the Sefiros. He goes on to explain that this exists in Mitzvos which is why accompanying the dead takes precedence over learning Torah, and other such examples.
For the purpose of definition, I think you could say that the partzufim, like the sefirot, are the self-manifestations of divinity, which, behind the manifestations, is hidden and completely unknowable.
In a few key places in the Zohar, instead of talking about the ten sefirot, the text talks about other entities, called partzufim. The Ari reconciled the two terminologies by saying that after an initial state which did not endure, the sefirot of what would become Atzilut were reorganized into the partzufim.
So, for people engaged in the teachings of the Ari, the partzufim become the primary subject of learning. What they are and what it is that is being learned when one learns these teachings is elusive and not necessarily directly communicable (and there are several major streams of interpretation that take different paths).