Bond With Him
Get to know your bird before you begin training with him. Observe his personality and mannerisms, bond with him, and form reasonable expectations. Do some research on your bird's species to figure out how likely he is to talk, and how much you can reasonably expect him to say.
Watch Your Mouth
When you're ready to start training your bird, choose short words to start off with. When you talk to your birds, say the words you've chosen to teach him with enthusiasm. Use a happy, positive tone. Watch your bird closely as you say these words, and gauge his reaction. Some words will likely catch his attention more than others. Choose the word your bird responds best to to be his first word. Make sure you watch the words you say around him, to avoid accidentally teaching him the wrong words.
Repeat Often
Once you've found the perfect word to teach your bird, repeat it very frequently. Most birds mimic through repetition, so repeating the word to your bird is the best way to teach it. You can choose to teach your bird yourself, or use tools like a tape recorder or CD to aid you. One on one interaction is best, however.
It might take your bird a while to learn, but don't get discouraged. Set up a training routine and work with your bird every day. If your bird isn't responding to training, try teaching him something a little easier, like whistling.
Give Praise
Be sure to praise and reward your bird if he does seem to be responding. If he says the word, praise him, and reward him with a treat. He'll be a little more enthusiastic about repeating the word in the future.
With these methods, your bird will be speaking in no time. Just keep in mind that not every bird is prone to talking. Do some research on your bird's breed to see how likely he is to talk. If he's not likely to talk, you can probably teach him other things, like whistling and making certain noises.