First, I'd like to thank my mom for teaching me to do this. I've met a few people who grew up with dish washers, and weren't really sure how to do this when they got their first cheap apartment without one. I thought I'd write this for other people in the same boat. If nothing else, it may save a few roommate relationships.

Now take a look at the picture. This is my setup. I have the drain rack on the counter, and keep the dirty dishes on one side of the divided sink. If you need to conserve water the drain rack should really be in the sink, to allow you to rise the whole rack of dishes at once. The drain rack I have doesn't fit in this sink, so I'm doing it this way.

Tools:
1. Dish Soap - Just about any brand works. You'll need to use a little more of some of the cheaper ones.
2. Drain rack - You can buy these in big box stores and hardware stores
3. Plastic net puff scrubber - These work pretty well, and dry quickly, but gunk tends to get stuck in them, especially cooked egg.
4. Dish Brush - These are good for knocking the big chunks off. They are great for pre-washing for the dish washer, or knocking things into the garbage disposal, but they don't really get the crusty bits off very well. If you want one, look for one that has a straight edge you can use to scrape with.
5. Dish cloth - This is the traditional tool. They are good for wiping down counters, and you can use them to wash dishes. They have some limitations. Because they are absorbent, germs grow on them if you aren't careful. You need to wash them out with soap after each use, and boil them, or put them in the microwave soaking wet to kill germs. DON'T MICROWAVE A DRY DISH TOWEL or SPONGE. You will start a fire. You should also hang them to dry between uses.
6. Sponge - I don't have one at the moment. These are a lot like dish cloths in use and maintenance, but they don't dry as fast.
7. Green scrubber pad - These are my favorite for dish washing. They scrub well, dry quickly, and don't get too much food stuck in them.

Step 1: Fill the Dishpan/sink

In these instructions, I'm using a double sink. If you don't have a double sink, that's not a problem. Instead of filling one side of the sink, use a plastic dish pan. If you don't have a plastic dish pan, use the biggest widest pot or bowl you have.

1. Wipe out the sink and rinse
2. Plug the drain. If you don't have a plug, go to the hardware store. This one came with the apartment. If you're buying, I'd go for the cheap flat rubber one that fits all sized drains.
3. Add some dish soap right under where the water is going to hit. About two teaspoons works with this soap and this size sink. The amount needed varies with the brand and the dish pan size. You're aiming for just enough soap to cause suds to cover the top of the water.
4. Fill the sink about 1/3 with bath temperature water. You want hot water, because hot water cuts oils.

Step 2: Clean to Dirty

You're going to be using the same tub of soap water to scrub the whole sink of dishes, so it's important to clean the cleanest things first. Generally this means glasses, dishes, pots and pans. You need to use a little common sense here. Sometimes the order will vary depending on what's been cooked. Oily dishes use the most soap power and should go last.

Step 3: Soak - Scrub - Rinse - Drain

1. Soak - Put several items into the soapy water, starting with the cleanest. Putting items into the water before you scrub them lets them soak for a bit, and makes the scrubbing easier. Don't overload the sink. you still want room to work.

2. Scrub - Pick up one thing at a time and scrub it all over. You should use two hands for this. The picture only shows one because I needed one hand for the camera. Friction is your friend.

3. Rinse - Once scrubbed, put the dish under the faucet on the dirty side if it will fit and give it a quick rinse with warm water.

4. Drain - Place it in the drain rack.

Step 4: Silverware Speed Trick

1. Put a big pot or a bowl in the middle of the sink as a place holder.
2. Put all your dirty silverware above in the water above the pot with the handles facing away from your scrubbing hand.
3. Grab about 2-3 utensils at a time by the handles, scrub them off with your other hand
4. Once scrubbed drop them back in the water below the pot facing the same way.
5. Once they are all scrubbed, grab them all by the handle and rinse them all at once.

Step 5: Special Treatment Items

There are a few things that need some special treatment

1. Non-stick Teflon coated pans - When you buy scrubbing tools, be sure they say Teflon safe on the package. If you're not sure, use a sponge, or dish cloth to avoid scratching the Teflon off.

2. Anything with a metal cutting edge - Don't ever drop these things into the soapy water. You can dunk them, but don't let go. This is so they don't rust, but also, so they don't cut your hands up.

3. Wooden cutting boards - These are prone to warping when left wet, or even on a wet surface. When they warp, they crack, or the seams come unglued. Don't soak them in water. You can wipe them off with soapy water, and put them under running water, but it's a good idea to wet both sides at once.

4. Cast Iron skillets - If it's not yours, ask the owner. People have very particular, and sometimes peculiar ideas about how to clean these. This is all related to not wanting to wear off any of the burnt on seasoning, which works as a non-stick surface. To avoid controversy I'm just going to direct you to the company that made mine. Link to Lodge cleaning instructions

5. Nice china with metal edges - Some nicer dishes have rims, or other decorations that are metallic. Use a soft dish cloth, or sponge on these, to avoid scrubbing off the metal decoration.

6. Blenders - Screw off the bottom of the blender to wash. If you don't gunk collects in the joint.

Step 6: Clean Up

1. Pull the plug on the sink and give it a quick rinse.
2. Empty out the drain strainer.

Step 7: Dry the Dishes - and Put Away

I've stopped doing this step. I just do the dishes in 1 drain rack batches and let them air dry. If you have a large family, or throw big parties, you may need to dry them to make room. To do this, you'll need a large thin cotton dish towel. Just pick a dish, dry it off and put it away. Don't use the dish drying towel for anything else. You want to keep it clean.