One of the parrot's most amazing abilities is their ability to not only mimic humans, but have an understanding of what words mean. African Gray parrots are known to be one of the best talkers, with the ability to learn thousands of words and even put them together to convey different needs or emotions.
Teach your parrot to trust you as the flock leader. Parrots will mimic the sounds of their flock in order to communicate, and as the flock leader you will be in charge of what sounds are used.
To become the flock leader you must earn your parrot's complete trust, and also trust your parrot. Keep your parrot's eye level below your own at all times to show your status as the "higher up."2
Talk to and around your parrot as often as possible. If you have specific words you would like your parrot to say, say them often. Use them in the context you want your parrot to use them in. (For example, say hello every time anyone enters the room if you would like your parrot to greet people.)
You should talk to your parrot every day, while he is in his cage and while he is out. The more often parrots hear words, the sooner they will begin to mimic them.3
Play recordings of words while you are gone. You can record tapes of yourself speaking, or buy prerecorded CDs.
Remember, though this is a good way to reinforce words you would like your parrot to use, it should not take the place of you speaking to your parrot yourself. A parrot has no reason to want to interact with a radio, and so they are very unlikely to respond to it.4
Listen to your parrot when you are out of the room. Some parrots become shy and practice words until they are confident using them in front of others. If you hear your parrot trying to say a word, say the word in response to encourage your parrot's efforts. Soon, your parrot will be confident enough to use the word in front of others.