All-Grain - is when the brewer creates the entire wort (the liquid before it becomes beer) by mashing crushed malted grain and running hot water through the grain bed in a process called lautering. All-grain is the holy grail of brewing, however, it does require more equipment as well as an additional commitment of time.
Partial/Mini Mash - this technique is for someone who wants the advantage of a mash but it requires much less time and equipment. The brewer uses a smaller amount (hence the word "mini") of grain and supplements the batch with malt extract. This method gives you the advantage of a mash but requires much less equipment and time.
Extract - malt extracts (liquid and/or dry) are used which allows you to skip the mashing process and go right to the boil and fermentation steps. In extract brewing, malt extract is added directly to the brew pot and boiled together with hops to create the wort. You can make very high-quality beer using extracts and is one of the most popular used by homebrewers. Many recipes and/or kits call for steeping grains prior to the boil. This is when the brewer takes a small amount of grain and steeps in hot water for 20-30 minutes to add fresh flavor, color and aroma.
No boil - extracts are available already hopped and require no boiling time. If you are in a hurry and want to "whip" up a batch of beer in less than an hour, this is a good alternative.