So to recap:
Ionic bonding - occurs when your compound contains BOTH metals and non-metals
Covalent bonding - occurs when you ONLY have non-metals present. This applies to both non-metal elements and compounds made of only non-metals
So the last scenario is what happens when we only have a pure metal! Well we know metals exist, they are all around us! So what happens when atoms and electrons when we only have a metal? Let's look at a really, really easy example.... the metal sodium
Sodium is in Group I, so its atoms only have one electron in their outer shell.
The problem is that every single sodium (Na) atom wants to lose one electron to achieve a full outer shell. It cannot share electrons with another sodium atom, as covalent bonding ONLY occurs with non-metals.
So what happens?
Well this is a little like ionic bonding.... the sodium atom loses its one outer shell electron.... but there isn't anything to 'pick it up' like there is in ionic bonding, so it just sort of floats around. Now every single sodium atom does this and you end up with something looking like this.....
Once the sodium atom loses its one outer shell electron it becomes a positive charged ion. They are represented by the big circles above with the + inside. Those positive ions then arrange themselves in huge three-dimensional layers, called a lattice!
Now that sounds like a bit of a problem.... two ions that both have the same electrical charge (positive in this case) should repel each other and fly apart. But why don't they?! Well this is the cool bit. Every positive sodium ion was formed when a sodium atom lost its one outer electron and all those electrons are now floating around in the gaps between the positive sodium ions. Because those electrons are negatively charged, they act like a glue when they pass between the positive ions and they hold the whole structure together.
We say the electrons are 'delocalised'. This means that you have no idea which delocalised electron belongs to which sodium ion.... they are always moving and zipping between the gaps!
This is also why metals are fantastic conductors of both electricity and heat. It is these delocalised electrons which are moving between the large stationary positive ions, which form an electrical current or transfer heat.