There are three main methods for brewing - All Grain, Partial Mash, and Extract. I am sure others will argue there additional ones, but for the sake of an arguement we'll stay with these three.
To brew beer, you need four ingredients at a minimum, fermentable sugars, yeast, hops and water. You can add more stuff if you want but those four make beer beer.
The three methods differ in how the fermentable sugar is derived. With All Grain, the sugars are essentially extracted from the grains. Partial Mash is a mix between getting the sugars from the grain as well as using some type of malt extract (liquid or dry). Finally, Extract is using concentrated malt extract as the base for the fermentable sugars.
At a high level the process for All Grain is
1) Soak the grains in water
2) Rinse the sugar off of the grains into a big pot
3) Boil the liquid with some hops
4) Cool, add yeast and let ferment
5) Drink
Partial Mash is essentially the same but you add some malt extract in step 3.
Extract eliminates steps 1 & 2 and uses all extract in step 3.
The entire process is more elaborate than these 5 steps but hopefully that help explains how the three methods relate to each other.
As far as time - Extract is the quickest, then Partial and then All Grain
As far as simplicity - For a beginner, extract is the easiest. I would not recommend using the other two until you have some experience. There are very experienced homebrewers who only brew extracts so quality is not sacrified for simplicity in this case.
As far as cost - The initial investment in equipment for Partial Mash or All Grain can be significantly more than extract. All three methods use the basic equipment, but all grain requires bigger pots than you may need for extract as well as some additional equipment to support steps 1 & 2 above. That being said, the ingredient cost to make a batch tends to be lower for all grain. I am sure some will debate that point though.
My advice is stick to extract, determine if you like it, if so, gradually buy or build what you need to advance. The nice aspect of this hobby is that it is not very linear to advance. You can stay with extract but move from bottling to kegging. You can move to all grain but still use bottles. You can make your own recipes, use ones from this site or buy kits. You can use the same kit and change one of the ingredients to get a different tast. There are so many variables that you can continually make changes gradually without investing a lot of money at one time.
I would also recommend the Palmer book as have others. Some of the major etailers also have introduction to brewing material on their sites.
http://www.midwestsupplies.com/faq?catid=1
http://www.austinhomebrew.com/videos.php
http://www.northernbrewer.com/brewing/marketing-categories/new-brewers/
http://morebeer.com/search/103579
If you do like it and stay with it, check the classified ads here as well as the DIY section to buy or build additional equipment instead of buying it new.
