Why do some balloons float upwards and others don't?

Every one loves balloons. Round, colourful and they explode with a bang.
Have you wondered why the balloons we blow ourselves don't float, while some float up and away?
Hold your breath...the secret is about to be revealed. 

Floating in general
We are all comfortable with things that float in water. We see that happen every day - the paper ships we send floating in rainwater and the plastic toys that keep us company in the bath tub. In fact, in swimming pools we have seen people float in water. The same reason why things float in water applies to air as well.
Let's say that you take a plastic 1-liter soda bottle, empty out the soft drink, put the cap back on it (so you have a sealed bottle full of air), tie a string around it like you would a balloon, and dive down to the bottom of a swimming pool with it. You can sit at the bottom of the pool with it, holding the string, and it will act just like a helium balloon does in air. If you let go of the string the bottle will quickly rise to the surface of the water.
The reason that this soda bottle "balloon" wants to rise in the water is because water is a fluid and the 1-liter bottle is displacing one liter of that fluid. In other words, because we introduced the bottle in the pool, the bottle pushed away some amount of water to make space for it.
The bottle and the air in it weigh perhaps an ounce at most (1 liter of air weighs about a gram, and the bottle is very light as well). The liter of water it displaces, however, weights about 1,000 grams (2.2 pounds or so). Because the weight of the bottle and its air is less than the weight of the water it displaces, the bottle floats. This is the Law of Buoyancy.
The Helium Balloons
Now that we understand why things float, let's get back to balloons. The reason why some balloons float while others don't is because of the presence of a gas called Helium. Balloons filled with Helium float in air, while the one with just air don't.

Now, Helium balloons work by the same law of buoyancy. In this case, the Helium balloon that you hold by a string is floating in a "pool" of air (when you stand underwater at the bottom of a swimming pool, you are standing in a "pool of water" maybe 10 feet deep -- when you stand in an open field you are standing at the bottom of a "pool of air" that is many miles deep). The Helium balloon displaces an amount of air (just like the empty bottle displaces an amount of water). As long as the helium plus the balloon is lighter than the air it displaces, the balloon will float in the air.