"Sea of Electrons"
Covalent solids, also called network solids, are solids that are held together by covalent bonds. As such, they have localized electrons (shared between the atoms) and the atoms are arranged in fixed geometries. Distortion away from this geometry can only occur through a breaking of covalent sigma bonds. As a result, the melting point of covalent solids is extremely high. They also tend to be extremely hard substances that will break into pieces rather than smoothly change shape. We say that they are stiff and brittle.
Examples of covalent solids include diamond and silica (SiO2).
Ionic solids are solids held together by ionic bonds. They are lattices composed of oppositely charged ions. Ionic solids tend to have high melting points ("high" generally being higher than metallic but lower than covalent). Because the electrons in ionic solids are localized (in the ions) these solids tend to be stiff and brittle like covalent solids.
As solids they do not conduct electricity (again because of the localize electrons). However, they do conduct electricity as liquids (or when dissolved in water) since then the ions can move.
Examples of ionic solids include NaCl and Ca(CO3). Ionic compounds are all generally found in nature as ionic solids.
"Sea of Electrons"
Metallic solids are solids composed of metal atoms that are held together by metallic bonds. These bonds are like huge molecular orbitals that span across the whole solid. This means the electrons in metallic solids are delocalized. They are not just held between a couple of atoms in a sigma bond. Rather, there is a sea of electrons everywhere. A good picture of a metal solid is that of cation in a sea of electrons. Because the electrons are delocalized, it is possible to move the nuclei (the cations) without a huge amount of energy. It is a bit like breaking a pi bond versus a sigma bond. As such, metals tend to be "soft." They are malleable which means they can be pounded into sheets. They are also ductile, which means they can be pulled into wires. The key idea is that unlike covalent solids, they are not stiff or brittle. That is because the cations in the sea of electrons can slide around without needing to break any very strong bonds.
Also, because the electrons are delocalized, metallic solids tend to be good conductors of electricity (moving electrons).
Examples of metallic solids include, copper, gold, zinc, .... We tend to think about metallic solids as pure metals, but they can also be combinations of metals like bronze, which is a mixture of copper and tin.
Molecular solids are solids that are essentially collections of molecules held together by intermolecular forces (IMFs). The solid structure is maintained by IMFs rather than bonds (metallic, covalent, or ionic). The forces holding the solids together are much weaker than for other types of solids. As a result these materials have much lower melting points. Molecular solids also have localized electrons (localized within the bonds in each molecule) and as such, do not conduct electricity.
Examples include ice (solid water), dry ice (solid CO2), solid iodine, and napthalene to name a few.