Animals that you find on the street can be quite dangerous, you must always remember that they are probably frightened by you, and if you unintentionally threaten them, they will become violent. Follow these steps and ideally, the animal will learn to trust you, and hopefully become tame enough for you to bring home.

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[edit] Steps

Get the animal's attention: Usually, making a soft clicking noise with your tongue, or by talking to it quietly will be effective. It is imperative that you do not startle the creature.
Approach the animal VERY slowly. Do what you can to appear as small as possible and hopefully less intimidating, but remember to stay attentive. Do not get down on your hands and knees, it will make it harder for you to get away if the animal decides to attack. Only come within 10-15 feet of the animal.
Allow the animal to make the final approach once you are close. Call to it in a soothing voice and try to get the animal to come to you. Put out your hand, gently pat the ground, and you can also toss the animal food, throw the food to the side and not directly at the wee beast.
Avoid prolonged eye contact. Look to one side of the dog rather than staring it down. Dogs consider this to be a sign of domination. They interpret it as a kind of challenge and it can cause the dog to think you want to fight. (Ever notice that dogs you know will usually not keep your gaze for long? Or cats that glare at each other for as long as possible before a fight. To them it's very rude, so this is why. And why it's best to avoid it!)
Hold out your hand so that the back of your hand is facing the animal. This is less threatening to most animals, and also lessens the chances of your fingers being bitten.
Observe the animal's body language if it does not approach. It's best to let cats come to you, they are hunters themselves but small enough to know the perils of being the hunted! But for dogs you may slowly - very slowly - start to take a few SMALL steps toward the animal if it seems friendly but just shy.
Stay calm and slowly back away if a dog and snarls or bares its teeth... or a cat and it's growling or hissing at you. Do NOT run.
Remain motionless as the animal sniffs at your hand.
Allow the animal to finish it's examination of your hand and then slowly move your hand from the front of its face to just behind one of its ears, and scratch or pet gently. Remember that many animals have areas that they do not like having touched, so go slowly.
Wait until the animal is comfortable before reaching for the ID tag. Speak softly and move slowly to avoid frightening or startling the animal.

[edit] Tips

Food is a wonderful way of winning an animal's trust. Start by gently tossing a small piece to one side of the animal, and not directly at it.
If it growls at you, don't worry or be afraid; animals can sense fear and know when you're afraid. Then they think they are the boss. Just stay calm and relaxed, it's usually just intimidated by you.
Never corner an animal. It will make it feel trapped and feel as if it has no option but to attack you.
A friendly cat will rub your hand after sniffing it for a moment.
Even after the animal has determined that you mean no harm, this does not mean that sudden movement will not scare the animal. Remember to move slowly.
If you cannot gain the trust of the animal, but fear for its life or safety, the best thing to do is call your local animal shelter. They may also advise you on how best to handle gaining the trust of the animal.

[edit] Warnings

If it's a dog and it snarls and bares its teeth or a cat and it's growling or hissing at you, do not run. Just stay calm and slowly back away. (restated here, as this is important!)
Be VERY VERY CAREFUL when approaching a strange animal! If it is stray, it may have not had any KIND human contact, for a while, and will not hesitate to attack you if it thinks you are dangerous! Even if it is domestic, it may be lost and desperate.
People are often discouraged to approach a strange animal, as there could be a rabies risk. However, it is likely that you will find that rabies is all but non-existant in your region. If you want a new furry friend, don't be afraid to befriend a stray. Just be SURE to take it slow, and when you have decided to keep your new friend, get them seen by a Vet asap.Or you can consider adopting from an animal shelter.
Make sure it doesn't have any illness and has been vaccinated properly.