In many articles, it has been mentioned that bulletproof vests also contain ceramic plates for extra protection. Wouldn't it be better to use metal instead? Since ceramic breaks easily.
I think Wikipedia provides a very good summary:
Unlike metals, ceramics are never used alone, as standalone materials, in armor systems; they are always combined with a ductile backing or support layer of metal or fiber reinforced plastic composite materials, and this ceramic-faced assembly is called ceramic armor. Ceramic materials, like glass, have high hardness and compressive strengths but low tensile strengths. Bonding a ceramic tile to a metallic or composite backing material, with high strength and good ductility, delays or mitigates tensile failure upon impact, and forces the ceramic to fail in compression.
So ceramic would never used on its own in a bulletproof vest. It will always have metal or something with similar strength.
Ceramic armor systems defeat small arms projectiles and kinetic energy penetrators by two main mechanisms: Shattering and erosion. When a hard steel or tungsten carbide projectile hits the ceramic layer of a ceramic armor system, it is momentarily arrested, in a phenomenon known as dwell. Depending on the thickness and hardness of the ceramic layer, the projectile core is then either shattered, fractured, or blunted. The projectile's remnants continue to penetrate the comminuted ceramic tile at a reduced velocity, which erodes those remnants and reduces their energy, length, and mass. The metal or fiber reinforced plastic composite layer behind the ceramic layer then arrests the projectile's fragments or its eroded remnant, and absorbs residual kinetic energy, typically via plastic deformation.
Even though ceramic breaks easily as you said, it has high compressive strength, i.e. it can resist compression to a large extent. When combined with metal backbone, this helps to reduce the kinetic energy of the bullet and thereby reduce the damage.
As an aside, I should add that this idea that bullets can only be stopped with a hard material like metal is not always true. While metals can stop bullets, they are also heavy and reduce mobility. Stopping bullets requires that its kinetic energy be absorbed and dispersed accross a larger area so that the impact is no longer felt. This can be done with materials other than metal. For example, during the Vietnam war, flak jackets (armor) without any metal plating was used. These would be made from multiple layers of ballistic nylon, along with fibreglass plates, and had much less weight than metal plated armors. The multiple layers of strong fabric and the fibreglass would spread the impact of incoming projectile across a larger area. Note, however, that these jackets were only effective against grenade fragment, shotgun pellets, and possibly small pistols. They would not work against rifles or anything stronger.
In the modern day, polymeric materials like Kevlar are used to protect against bullets. There are also many researchers working in the material science side of chemistry to find new materials that can be light but withstand impact.