A simple view of double covalent bonds
A double covalent bond is where two pairs of electrons are shared between the atoms rather than just one pair.
Some simple molecules containing double bonds
Oxygen, O2
Two oxygen atoms can both achieve stable structures by sharing two pairs of electrons as in the diagram.

The double bond is shown conventionally by two lines joining the atoms. Each line represents one pair of shared electrons.
Carbon dioxide, CO2

Ethene, C2H4
Ethene has a double bond between the two carbon atoms.

A more sophisticated view of the bonding in ethene
It is important to explore the bonding in ethene in more detail because it has a direct impact on its chemistry. Unless you have some understanding of the true nature of the double bond, you can't really understand the way that ethene behaves.
An orbital view of the bonding in ethene
Ethene is built from hydrogen atoms (1s1) and carbon atoms (1s22s22px12py1).
The carbon atom doesn't have enough unpaired electrons to form the required number of bonds, so it needs to promote one of the 2s2 pair into the empty 2pz orbital. This is exactly the same as happens whenever carbon forms bonds - whatever else it ends up joined to.

Now there's a difference, because each carbon is only joining to three other atoms (another carbon and two hydrogens) rather than four - as, for example, in methane.
The carbon re-organises the s orbital and two of the p orbitals to give three new orbitals with exactly the same energy. This process is called hybridisation. The other p orbital is left unchanged.