There is no one right way to transplant your house plants, and different plants may benefit from slightly different approaches. This is a very generic explanation that will work well for most any plant that have out grown their containers.
Choosing a New Pot
When you've decided your plant has outgrown its pot, the first step is to find an appropriate replacement. A good general rule of thumb is that the new container should be at least twice as big as the old one. Being transplanted is traumatic for a plant, so you don't want to have to do it all the time. It's also easier to fill the new pot with the right materials if you have some space around the root mass to work with. Hold the old container above the new one to get a feel for it. The below example is just barely big enough. If possible, go bigger. The new pot should have good drainage. If the drainage is lacking, it should be all that much bigger.
A new pot that's just big enough for the plant
Preparing the new pot
Before you start filling it with dirt, take some time to prepare the new pot.
1. Clean the pot if it's dirty.
It doesn't need to be squeaky clean, especially on the inside.
2. Protect the drainage
Locate the drainage holes and put some material over and around them to prevent them getting clogged. The fewer drainage holes the more important this is. In this example, there is only one small drainage hole, so we'll be very diligent. I used clay pellets and moss, but course gravel works well too.
Carefully place some material such as moss or larger stones near the hole(s)
Continue to fill in around the holes with more material
Too much is better than too little. You don't want that hole to get clogged.
3. Check your spacing and add some dirt
Hold the old pot inside the new pot to get a feel for how much dirt to put in ahead of time. Most plants do best if they are a little high up (the top of the root ball can be a little above the soil). Then add some dirt to the bottom of the pot and try holding the old pot in the new one again to see if it's enough. Repeat until you feel like the plant will be held high enough off the pot bottom by the new dirt. At this point, your new pot should have some dirt in the bottom like this:
A pot with dirt in the bottom, ready for a new plant.
Transplant the plant
The plant must now be extracted from its old container and the roots have to be prepared for being transplanted.
1. Loosen up the roots and dirt, then extract the plant
If your plant is in a plastic or other flexible container, you should rotate the container while squeezing the sides to loosen it up as shown in this picture:
If the old container is hard, you'll need to use a tool such as a small, hard stick, a knife, spoon, fork, or spade to loosen the dirt.
Once it's loosened up, try to extract the plant by carefully tipping it upside down while protecting it from falling on the ground. Coax the plant out by shaking lightly, tugging lightly, or continuing to work at it with the tool (fork, stick, whatever)
2. Prepare the root ball
This is especially important if you've allowed the plant to become root bound like this one. Notice how the roots have formed a solid mesh. This will prevent the roots from easily growing into their new medium. To solve this, very gently break apart the root ball until the roots are more loosely exposed. When you're done it should look more like this:
3. Place the plant in its new home
Holding the plant in place, carefully pack soil around it in the new pot, going around and around the sides and pressing and filling until you've filled it close to the top. Be sure to leave some space at the top of the container for watering and feel free to add some decorative flair if the dirt is exposed.
4. Watch and water...
For the first week or two your plant will be in shock. It will need bright light without too much heat, and plenty of water. Don't use any chemical fertilizers right away. Allow your plant at least a month in its new home before using fertilizers.