What property of objects allow them to float? I used to think that the shape of an object determines its ability to float (boat-shaped objects are more likely to float, and spheres tend to sink).  But my friend, who is fond of making me look stupid, took me to the local lake showed me a sphere that floated and a boat-shaped object made out of iron that sunk.
Is it based on the mass?  I'm not sure that is possible, because I've seen really heavy things (like airplane carriers) float, and really light things (like my friend's iron boat) sink.
What property of certain objects allow them to float, if any?
 A: Actually, the answer is a bit more subtle than just density. The principle that is behind floating objects is Archimedes' principle:

A fluid (liquid or gas) exerts a buoyant force, opposite apparent gravity (i.e. gravity + acceleration of fluid) on an immersed object that is equal to the weight of the displaced fluid.

Thus, if you have an object fully immersed in a fluid, the total force it feels is given by (positive sign means down):
$$F = \text{gravity} + \text{buoyancy} \\= \rho_\text{object} V g - \rho_\text{fluid} V g \\= (\rho_\text{object} - \rho_\text{fluid}) V g$$ 
Thus, if the average density of the object is lower than that of the water, it floats. If the object is partially immersed, to calculate the buoyant force you have to consider just the immersed volume and its average density: 
$$F = \rho_\text{object} V g -  \rho_\text{fluid} V_\text{immersed} g$$ 
Note that when I was talking about density, I was talking about the average density of the object. That is its total mass divided by its volume. Thus, a ship, even if it is made out of high-density iron it is full of air. That air will lower the average density, as it will increase the volume considerably while keeping the weight almost constant.
If you want to understand this better you can give the following problem a try :) 

What is the height an ice cube of side L floats in water?

A: DENSITY
It is because of densities of the object that is floating and the liquid in which it is floating.
If an object have density lower than a fluid it will float otherwise it will sink.

Density of entire object [mass / volume] should be
  taken into account and not merely the
  density of material it is made up of.



*

*A ship made up of iron floats in sea because density of ship i.e. mass of ship/ volume of ship is less than that of water. Here though density of iron is more than water hollowness of ship makes its volume large hence density of "ship" is lower than that of water. 

*In dead sea you and I can float.

*In mercury iron nails float.
A: An object floats if its upthrust (buoyancy) is in equilibrium with its downwards gravitational force.
In other words (as stated by the wiki page),
$$F_net = 0 = mg - \rho_f V_{disp} g$$
(where all the constants are pretty self-explanatory.)
Clearly then, the properties of the object that determine whether/how it floats are its mass and volume. More specifically, it is the relationship between the two; the density of the object (considering the enclosed volume, i.e. that which water can't enter), must be lower than the density of water $\rho_f$ for it to float.
Note also that is the density function (how the density of the object varies spatially) that determines what  and the shape of the object that determines exactly how the object floats (e.g. the water line). This is not necessary just to determine if it floats, however.
A: The only parameter that matters is the density: the ratio between the weight and the volume.
If the density is higher than the surrounding medium (let's say water, whose density 1000 kg/m³) is will sink and it will float if the density is lower (you have the special where they are both equal).
Objects float due to Archimedes' principle: basically, and under some hypothesis, there is a force, directed upward, equal in magnitude to the weight of the volume of displaced water (the volume of water that the immersed part of the body occupies). This weight is equal to g * V * density . The other force acting on the object is its own weight: g * V * density object .
You then recover the rule about densities.
A: In the most common cases, it's based on density. An object that is denser than the liquid will sink, and one that is less dense will float.
To go into a little more detail: when an object is placed in a liquid, the liquid exerts an upward force on it, called the buoyant force. This force is equal in magnitude to the weight of the liquid displaced by the object - that is, the amount of liquid that would fill the submerged volume of the object. (I can provide more mathematical detail if you like, or just look at the Wikipedia article linked)
Now, suppose the object is completely submerged.


*

*If the object is denser than the liquid, the weight of the liquid that would fill the object's volume will be less than the weight of the object. So the buoyant force, acting upward, is not strong enough to counteract the object's weight, and the object sinks.

*If the liquid is denser than the object, the weight of the liquid that would fill the object's volume is more than the weight of the object, and the buoyant force is strong enough to overpower the weight, lifting the object up. But as the object rises, eventually it starts to rise out of the liquid, its submerged volume becomes less, and in turn the buoyant force becomes less. At some point, the buoyant force will drop to be equal to the weight of the object, and then the object will stop rising - it'll just float there, at that level.
A: An object floats when it receives enough push from the water to compensate its weight. The push is equal to the weight of the water displaced.
In other words: objects with a particularly good shape for moving water away (like boat-shaped objects) are allowed to weigh more and still float; objects with a bad shape (like spheres) are allowed to weigh less and still float.
A: To solve any mechanical problem like this, you need only consider the forces acting on an object.  In this case, there are two relevant forces on the boat: its weight (pulling it down), and the buoyant force (pushing it up).  The buoyant force on a boat is equal to the weight of the water it displaces (see: Archimedes principle).  As the boat sinks into the water further, more and more water is displaced, until the weight of the water displaced equals the weight of the boat and equilibrium is achieved, and the boat floats at that level.
A: There are no properties of an object that makes one float and another not.  Whether something floats or not is determined by comparing its weight in air to the weight of the water it displaces.  If the displaced water weighs more, the object floats.
The water that would fill the boat shaped hole in the water weighs more than the boat itself.  That displaced water was raised up above its normal resting place and let's say all that raised water weighs 1000#.  Let's say the boat weighs 600#.  1000 pounds of elevated water wants to fall back down, and level itself, ejecting that boat (no matter how it's shaped).  So there's a 1000 pounds pushing up against the boat, and 600 pounds of boat pushing down.  The boat loses the battle and floats. 
Let's say a bowling ball could be filled with 1 gallon of water if it were hollow, but the ball is normal, and weighs 15 pounds.  When you put the ball in water and 8.3# (1 gallon) of water is raised in the lake, ocean, or bucket.  But the ball outweighs the displaced water by 6.7#, and still sinks leaving the water raised.  (The ball would weigh 6.7# underwater because the force of 8.3# water would push upwards offsetting the downward force of 15#.)  
Because water expands when frozen, ice takes up more space than the water it consists of takes up.  Therefore the water displaced by the ice weighs more than the ice, and the ice floats.
As you see, the reason anything floats is ultimately gravity.  The gravity of the displaced fluid trying to fall back down and reclaim its original space.  There is no such thing as "buoyancy", even though I've heard MIT professors use the term.
A: strong text  Take the weight (mass) of the object and divide it by the volume of it.  If the number you get is less than one, the thing will float.  If the number is more than one, it sinks. 
Take a copper sheet 8" by 8" of any thickness.  Drop it into water and it will sink.  Take it out and make a bowl of out of it.  (Beat it with a ball hammer in the centre while it is placed on soft ground (dirt) to make the bowl).  Now gently lower the bowl onto water.  It will float.
