Well, to understand the answer to this question, we need to understand what "heat" is. The general idea of "heat" is actually pretty complicated, but for now, we can talk about it this way: all the stuff in our everyday life is made up of atoms. These atoms are constantly vibrating and jostling around, and "heat" is the energy contained in this vibrating. So, the hotter something is, the more strongly the atoms it's made of are moving around.

Now we can talk about heat transfer. One method of heat transfer is conduction: let's say we have something really hot (so its atoms are vibrating really fast) and touch it to something colder (whose atoms are vibrating more slowly). Then as the atoms in the hotter object vibrate, they'll hit the atoms in the colder object, and make them start vibrating more. As this happens, the hotter object cools down, and the cooler object warms up. This is how, say, water in a pot on a stove absorbs heat: the stove heats up the bottom of the pan, and that heat is conducted into the water in the pot.

Another form of heat transfer is radiation. You may know that atoms are made of electrons (and other stuff), and that electrons create electric fields. Well, as the atoms in something vibrate, the electrons vibrate too, and this vibration creates waves of what's called electromagnetic radiation. Depending on how quickly the atoms are vibrating (so how how something is), the electromagnetic radiation can be what we know as radio waves, microwaves, visible light, X-rays, and so on. Now, if this radiation hits some other stuff, the reverse process can happen: the electromagnetic radiation hits the electrons in some material, and that makes the electrons vibrate, which makes the atoms vibrate, which makes the material hotter. This is how the radiation that the sun emits can heat up the Earth, even though the Earth and the sun are really far apart! This is also another way that water can absorb heat.

First, we need a molecular picture of what temperature means:
Water (similar to all substances) is composed of tiny molecules, like billiard balls on a pool table. When the molecules have a lot of energy, they bounce around at high speeds and collide with slower-moving objects around them, transferring some of their energy. At large lengths, this would be water vapor. When the molecules are moving very slowly, attractive forces between them pull molecules together into a tight crystal shape. This state is ice. In between these two extremes (at room temperature), liquid water molecules move slow enough to feel attracted to nearby molecules, but fast enough to be able to easily move around.

To answer your question, you can think of water absorbing heat like the break in a game of pool. A fast moving molecule (the cue ball) strikes a lattice of slow moving ice or water molecules. The cue ball rapidly decelerates (is cooled) as the rest of the balls jumble around the board in a liquid or gas-like state.

Water absorbs heat by vibrating and rotating. The vibrations can be the stretching or bending of the chemical bonds between hydrogen and oxygen.

This a good question, and it's related to the concept of thermal equilibrium.

Thermal equilibrium means that when we have two bodies in contact with one another, they will exchange energy until their temperatures are the same. in this example, we can think of air and water as the two bodies that are in contact and exchanging energy.

Water can absorb a very large amount of heat and only a small increase in its overall temperature. This property of water, its high heat capacity, is part of what allows life as we know it to exist on this planet by regulating global climate. Water has a high heat capacity (an ability to absorb heat) because for water to increase in temperature, water molecules must be made to move faster within the water; doing this requires breaking hydrogen bonds (the H₂ in H₂O) and the breaking of hydrogen bonds absorbs heat. With a such a high heat capacity, a lot of heat energy can enter a body of water before the water actually increases in temperature. This is why it takes water so long to boil!

Matter of any kind absorbs heat. Water is no exception.

Water is particularly good at it because of the way that hydrogen atoms in one molecule can form temporary bonds with oxygen atoms of another molecule. The process is called "hydrogen bonding". Most of the bizarre properties of water stem from this fact (e.g. liquid at room conditions, solid form [ice] is less dense than liquid form, and can absorb a tremendous amount of energy without warming up very much).

An object absorbs heat in a number of different ways. One way is by radiation... so, when you shine a light (Sun! on a rock, the photons making up the rays of light impinge on the atoms in the rock. This causes the atoms in the rock to VIBRATE faster because they have more energy (photon energy converted to vibrational energy).The higher rate of vibration is what we sense as HEAT.