Metallic bonding is one of the main types of strong chemical bond. As you might have guessed from the name, it's the kind that holds metals together. It's sometimes considered a form of covalent bonding, in the sense that it involves the sharing of electrons between atoms, but it has a few major differences from the kind of covalent bonding that holds non-metals together. For one thing, where the bonds in non-metals involve electrons fairly tightly bound between two atoms, the valence electrons in metals are free to move all around them. That's why metals conduct electricity; it's also why they form big lattice structures in their solid state, rather than small well-defined molecules.
We can think of the structure of a metal as a collection of positively charged atoms (cations) arranged in a regular pattern, with a sea of electrons swimming around them, sticking them together like glue. They fall into a regular pattern - a crystal - because that's the most efficient way of using the available space. You might notice the same thing with the way bubbles arrange themselves on the surface of a liquid when they arrive in a steady stream, or the way oranges arrange themselves in a box.
Despite the efficient use of space in a regular lattice, there is no guarantee that the whole metal will end up like that - if different regions start to crystallise at the same time while it's hardening, they'll probably be at slightly different angles. The bits where the differently aligned regions meet are called grain boundaries, and they're weak points in the structure. To make high-performance parts for turbine blades and so on, metals are very carefully grown as single crystals. However, despite the weaker bonds at grain boundaries, they can increase the overall strength of the material. Sometimes, small-scale failures can protect a structure from larger-scale failures, and irregularities can deflect stresses. Relatedly, mixing two metals together to make an alloy will usually lead to a harder metal, because it's harder for differently-sized atoms to slip past each other.
Unlike ionic compounds, which also form crystals, metals are generally malleable and ductile, rather than brittle - they can be hammered into shape, and drawn out into wires. The difference is that ionic bonding works with positively and negatively charged ions, which only attract each other as long as the positive ones are next to negative ones; as soon as a whack lines up positive with positive, that attraction changes to repulsion and the structure pushes itself apart. In a metal, all the ions are positive, and the negatively-charged electrons keep on sticking them together however bent out of shape they get.