All objects that we can see on and from Earth are composed of regular, everyday matter. This matter is composed of atoms, which are composed of particles; protons, neutrons, electrons and the like. Similarly, antimatter is composed of antiparticles; such as positrons, antiprotons and antineutrons. These antiparticles are, of course, composed of antiquarks. Antiparticles can even collect together to form antiatoms! Thus, all of our matter-composed elements are possible with antimatter - antihydrogen for example.
In an antiparticle, charge must be opposite, and mass must be basically exactly the same. Electrically neutral particles aren’t identical to their anti-counterparts, since they are still composed of antiquarks and antiparticles.
In 1928, Paul Dirac paved the first path to antimatter when he predicted positrons. Antimatter is not just a theoretical mathematical anomaly - it exists in nature. Antiparticles are created during beta decay, and in the interaction of cosmic rays (most notably gamma rays) and Earth’s atmosphere. Due to our universe’s conservation of charge, it is not possible to create an antiparticle without creating a particle of opposite charge or destroying a particle of the same charge. However, some (typically near or exactly massless) particles are their own antiparticles, such as photons, the theorized gravitons and some WIMPs.
Interestingly, when matter and antimatter collide - annihilation occurs. The collision can produce such emissions as high-energy photons (gamma rays,) or even other particle-antiparticle pairs. The particles that are their own antiparticles, such as gravitons and photons, can even annihilate with themselves! One of the greatest mysteries in Physics today is that, since this collision occurs, why the universe seems to be composed of mostly matter. In a process called baryogenesis, an asymmetry has occurred in the universe between matter and antimatter - and scientists are working hard as we speak to figure out why that is.