It's any batch from which part of its fermentables are derived from mashed grains and part from malt extract. It's like doing a small all grain batch to which you add malt extract and topup water to.
Mini-mash is just another term for Partial mash, if I'm not mistaken.
Like Bobby says, my understanding is that mini-mash is just like a baby all-grain recipe. It's got more grain than an extract recipe and more of the sugars are derived from the grain.
And dale beat me to the link while I was typing. That's a good one.
And it's well worth the effort. I have been partial mash from about batch #6 on and I'm at 35 batches now. I don't have the gear to go all grain but I get great results PMing using a converted 5 gallon round cooler for an MLT.