The interesting thing about molecules is that they do not necessarily have the same properties as the elements from which they are made.

This principle applies to the state of matter (gas, liquid, solid) and also many other properties. The state of matter of a molecule is determined by intermolecular interactions - interactions between the molecules. If the molecules strongly attract one another, they can stick together. If molecules stick together a little bit, they can form a liquid and if the stick together a lot, they will form a solid. There are several types of intermolecular interactions that cause molecules to be attracted to each other.

London Dispersion Forces.
The London dispersion force is a type of Van Der Waals force. These interactions are weak and they are proportional to the size of the atom or molecule. For example, hydrocarbon chains are made up of carbon and hydrogen. Carbon is solid at room temperature and hydrogen is a gas. The smallest hydrocarbon, methane (CH₄) is a gas at room temperature and pressure. So are the three next largest straight-chain hydrocarbons, ethane (C₂H₆), propane (C₃H₈), and butane (C₄H₁₀)).

However, the larger hydrocarbons, pentane (C₅H₁₂), hexane (C₆H₁₄), etc are liquids at room temperature because they are large enough for the London dispersion force to be strong enough to hold them together. For the forces to be strong enough to make the molecules solid, the hydrocarbons must have at least 17 carbons. The London dispersion forces between two hydrogen or oxygen molecules are not strong enough to cause them to form a liquid. A water molecule is bigger than a hydrogen molecule (has a higher molecular weight), so the London dispersion force is stronger; however, the molecular weight of water is smaller than that of oxygen, so London dispersion forces do not tell the whole story.

Dipole interactions and Hydrogen bonding
Dipole interactions and Hydrogen bonding are the second part of the explanation. Dipole interactions are Coulombic attractions between partially positive and negative charges on molecules.

When molecules are made up of elements with different electronegativities*, some of the atoms have higher electron densities than others, which causes some atoms to have partial positive charges and some to have partial negative charges. Oxygen is more electronegative than hydrogen, so in water the oxygen has a partially negative charge and the hydrogens have a partially positive charge. These charges cause the oxygen of one water molecule to be attracted to the hydrogen of another molecule. This interaction is much stronger than London dispersion forces. In fact, the interaction in water is called hydrogen bonding which is a particularly strong dipole interaction.

Hydrogen bonding is strong enough to cause even a small molecule like water to be a liquid at room temperature and pressure. Hydrogen bonding occurs between molecules with hydrogen bonded to fluorine, oxygen, or nitrogen and is strong enough that it is generally described as a separate category of intermolecular interactions from dipole interactions.

Therefore, the simple answer to your questions is that water has an extra type of intermolecular force at work, hydrogen bonding, which comes from the strong electric dipole on the water molecule caused by the electronegativity difference between oxygen and hydrogen.

If you take chemistry in high school you will learn about other types of intermolecular interactions including ion-dipole forces, dipole-induced-dipole interactions, and Coulombic attractions, etc.

*Electronegativity is a measure of the tendency of an atom to attract electrons when bonded to another atom. The higher the electronegativity, the more an atom attracts electrons in its direction in a bond. Fluorine (3.98), oyxgen (3.44), and Nitrogen (3.04) have the highest electronegativities.

References:
Lide, David R. CRC Handbook of Chemistry and Physics. Boca Raton: CRC, 2006. Print.