There are 3 main types of bonding: ionic, covalent and metallic.
Ionic bonds are are formed between metals and non-metals. Bonding takes place when a metal atom donates one or more electrons to a non-metal atom, resulting in them both having a full outer electron shell and they both become ions. Now they are oppositely charged ions, and their electrostatic attraction sticks them together making an ionic bond.
Ionic compounds are arranged in regular patterns. They have high melting points, are soluble in water. They also only conduct electricity while they are liquid (molten or in a solution).
Covalent bonding is between non-metal atoms. Non-metals want to gain electrons to fill their outer electron shell. They do this by sharing electrons, with their outer shells overlapping, allowing them to all have full outer shells. They can be split into 2 groups: giant covalent structures and simple covalent structures.
Giant covalent structures are made up of lots of atoms which are all covalently bonded together. They are very strong because of the strong covalent bonds and have high melting points, are insoluble in water and do not conduct electricity (with the exception of graphite).
Simple covalent structures have strong covalent bond within the molecules, but the bonding between molecules is very weak. This is because there is only weak intermolecular forces between the molecules. They have low melting points and do not conduct electricity.
Metallic bonding is bonding between metal atoms. The metal atoms give up their outer electrons to become positive ions. These outer electrons join together to make a delocalised sea of electrons. They are free to drug about the metal structure. Their negative charge attracts the positive metal ions that they surround so the metal ions stay in place.
Metals are very strong and have high melting points, can conduct electricity and heat and are malleable, which means they can be hammered out into different shapes.