Diagram of a molecule of water
In covalent bonding, the atoms are unstable because their outer rings of electrons aren't filled up. By sharing electrons with other atoms, these atoms can fill up their outer rings and become stable. In water, for instance, the oxygen atom needs two more electrons to be stable, and the hydrogen atoms each need one. When they get together, the oxygen atom shares one electron with each of the hydrogen atoms, and the hydrogen atoms each share one electron with the oxygen atom.
Now that the atoms have become stable, it's pretty hard to knock them back into being unstable again, so covalent bonds are strong and molecules that form with covalent (sharing) bonds are strong molecules.
Covalent bonding makes very strong connections between the atoms, so it's hard to break these molecules apart. On the other hand, molecules that join with covalent bonds aren't very much attracted to each other (unlike with ionic bonding), so they move freely around each other. That means that most molecules that form covalent bonds make either liquids or gases, like water and carbon dioxide. The main exception is metals, which hold together using covalent bonding but are still solids. That's why metals are so flexible and easy to melt so you can make them into different shapes.