They've updated the bricks and pieces system so my screen shots may not be as helpful as they once would have. I can't believe the horrible timing for my tutorial and their update.
Only recently (less than a year ago) did I discover "Bricks and Pieces." This is not to be confused with the online "Pick A Brick" (PAB) option. I had ordered parts many times from Pick A Brick when they had parts that I was looking for. I like PAB but I have found that "Bricks and Pieces" has lots of parts available to order that PAB does not offer.
For those that are NOT familiar with "Bricks and Pieces" I've prepared a tutorial to help. I hope that it helps others part with their money (its so easy to spend far too much)

Step 1: Go to LEGO shop at home (online shop on LEGO's website)
Step 2: scroll down the page (near the bottom and close to "quick links") and click "MISSING PARTS" Step 3: Enter age and location shipping to Step 4: Click "I want to buy a piece" Step 5: Click "Bricks and Pieces selection" Step 6: search by the set number OR element or design ID
This is where the difficulty begins. I like to open a notebook and assign different pages for different types of parts. For instance I may compile a list of design ID (or element ID) for all heads that interest me. On a second page I may compile a list of hats and hairpieces that interest me. On a 3rd I may compile a list of rare parts that interest me (printed parts even).
By making a list of parts and their design ID it makes it so much easier to find the parts later.
I made the mistake of just adding parts to my bag as I browsed and then later discovered that it was bedtime before I finished my order. Making a list makes it easier add the items later to your bag when you are ready to check out.
I think most people that go to Bricks and Pieces already have certain parts (or at least types of parts) that they are looking for. For example I keep hoping to find this pairpiece. The hairpiece of Clouse in Ninjago set 70748 and Kanan in Star Wars Rebels set "The Ghost."
At one point the part was listed when searching by set number (in both of these sets) but now it is not listed in these sets. Not all parts in particular sets are listed. Many are listed but are (in red) NOT available (out of stock). At one point this part was shown as "out of stock." Currently it doesn't show up at all in these sets in "Bricks and Pieces." Sometimes parts show up as "out of stock" and later are made available.
So I most often search by set numbers and make a list of parts that interest me. Be sure to notice that when you search by set to click the "more" button at the bottom to pull up more parts available in the set. Step 7: Write down the Element ID OR Design ID of parts that interest you (your bag will not be saved before checkout... so if you exit and come back your bag is usually empty).
Element ID is very precise and when entered will result in the exact type AND COLOR of the part.
Clicking on the part will pull up a larger picture of the part and show both the Design ID and Element ID.
Design ID will pull up the part in various colors (if they share the same Design ID). I most often only write down the Design ID so that I can browse available colors (especially if I may later choose a different color or order the same part in various colors). All of the axe heads shown in the screenshot below have the same Design ID... but a unique Element ID. So entering the Design ID will pull up what you see in the screenshot (one can then choose from various colors). In regards to torso's I like to record the Element ID (instead of Design ID) because entering the Design ID will result in LOTS of different torso's. Most parts don't have this many options for the same Design ID. Try it. Search for Design ID "76382". There are so very many torso's listed which is great news for those that want to browse but frustrating for those that have already made a list of items that want and accidentally write down the Design ID for a particular torso instead of the Element ID. Just to be clear I most often write down the Design ID (with the exception of torso's in which case I write the Element ID).
Step 8: Add the items that interest you to the bag.
After your have added an item to your bag it is difficult to determine what color you chose if there are other similar colors. This was especially frustrating for me in the regards to items such as weapons and armor that are available in light grey, dark stone gray, titan metal and black. When viewing the pictures of items in your bag it can be difficult to determine what color you have chosen. Check out the items in my bag in the picture. The helmet is black while the sword is silver metallic but in the pictures they appear very similar. Step 9: Review your bag to ensure that you have the right items and quantities.
Step 10: Checkout
I've had a lot of difficulty checking out. The last two times that I made purchases from "Bricks and Pieces" I spent hours browsing and searching for parts to complete my order only to have problems with the checkout process. This is very likely a result of trying to purchase far too many items. People that place small orders may not have this problem. Both times after clicking "checkout" my browser froze or instead of taking me to the checkout screen I was redirected to the homepage for LEGO shop at home. Once I was lucky and was able to click my browser "back" button and my items were still in my bag. Once I refreshed and my bag was empty. I had to enter the Design or Element ID's for each item again (start all over). I was so happy that I had made a list of parts so that I could easily find everything again.
Bricks and Pieces Pro's:
-Option of buying parts that interest you instead of buying the whole set or minifigure
-May be cheaper than ordering from bricklink (sometimes... other times bricklink may be cheaper).
-Better selection than "Pick A Brick"
Bricks and Pieces Con's:
-Checkout process can be brutal if technical problems occur (bag emptied). This may be the fault of my browser and not the LEGO site? I most often use Mozilla Firefox.
-Shipping fee's (not free for orders over $75 like most regular orders on LEGO [email protected])
Long delivery times. A order once took 4 weeks to make it to me. Keep in mind this is a custom order (lots of parts to locate for your order) and usually ships from a different location than ordering other items from LEGO [email protected]