Many extract kits come with some grains, but these are 'steeping grains'. These grains just add some color, mouthfeel, and a little bit of flavor to the beer...but NO fermentable sugar.
you just soak em in 150 degree water for 20-30 minutes, remove, then proceed with a normal extract boil.
a mini-mash is slightly more advanced. these grains actually contain the enzymes to extract fermentable sugars. The temperature of the water is very crucial for extracting the right sugars and avoiding tannin extraction (which will ruin your beer). you also need to think about pH.
these grains are typically 'mashed' in hot water for up to 60 minutes, then the wort is drained off.
you then add extract to finish reachign your target starting gravity.
a mini-mash is just like doing all grain brewing, except you only boil a couple gallons instead of the entire 5+ gallon batch, and all grain brewers get all their gravity from grain, without having to 'top off' with extract.
mini-mash is the same as a partial mash, which you can do a search for to get more detailed info.
Primary: English Mild
On tap: Pale Ale, Lancelot's Wheat, English Brown Ale, Steam Beer, HoovNuts IPA
Bottled: MOAM, Braggot, Raspberry Melomel, Merlot, Apfelwein, Pyment, Sweet mead, Cabernet
Gal in 2009: 27, Gal in 2010: 34, Gal in 2011: 13, Gal in 2012: 10