As you probably assume, producing antimatter is very difficult. Even understanding what antimatter is quite difficult. The simplest way to describe antimatter is that it is the "opposite" of matter. In general, we think that protons and electrons are opposite, but really they are not. They are all make of quarks, which therefore makes them matter. Antiquarks are the opposite of quarks--they have opposite charges (+/-, but also a "color" charge. This isn't really a color, just a mathematical way to describe an additional charge that we can't observe without very advanced instruments). At this level, everything is described by math, and may not make immediate sense outside of a math context. Essentially, an antiquark is a negative quark. And an antiparticle is a negative particle (made of antiquarks).

To make antimatter, you usually need a particle accelerator. These are huge vacuum chambers in the shape of donuts with diameters of miles. Using magnets, the article accelerator takes a proton or electron or other charged particle and speeds it up to ridiculously high speeds. It then smashes the electron (in this example) into a "target" made of atoms with high atomic mass (i.e. lots of protons and electrons). When these atoms are hit, they release packets of energy. Because there is so many photons are released in these collisions, some of it turns into electron-positron pairs (matter is not created, since +electron -electron = 0). Isolating this antimatter can be done with magnets to separate the charges, but it is very difficult.