mashing = soaking crushed grain (usually barley) in water at a certain temperature. Enzymes in the grain convert starches to sugars, which dissolve into the water, then you remove hte water and rinse more water through to get more sugars.
extract= manufactured malt sugars, either powdered or liquid (kits are extract with hop extract added)
all-grain= making beer using grain only, and no extracts. May or may not use adjuncts (other fermentables ie. sugar, honey, candi sugar, fruit etc)