How is antimatter made? How is antimatter made in laboratory? Can anyone explain, at the particle level, specifically how anti-protons and anti-electrons are made?
 A: Since John is not addressing positrons one should know that positrons are easily created once a photon has more energy than twice the mass of the electron, in electron positron pairs.
This can be seen clearly in this bubble chamber picture:
 
where the positron  is shown in purple on the right. One knows they are electrons (positrons) because of the rapid energy loss of the track due to the ionization, indicating the small electron mass.
To get positron beams one has to generate the pair on thin targets in vacuum, as the pair creation happens in the electromagnetic field of the nucleus, and then separate with magnetic fields the positrons from the electrons.
A: Creating anti-protons is straightforward in principle because any high energy collision produces a shower of protons, antiprotons and various types of pions. The pions decay in a few nanoseconds, so you just have to wait for the pions to decay then separate the antiprotons from the protons.
At Fermilab a 120GeV proton beam was collided with a nickel target to produce the protons and anti-protons. The proton/antiproton mixture was first passed through a lithium lens to produce a collimated beam, and subsequent magnetic separation produced separate beams of protons and antiprotons.
Although making antiprotons is easy, if by making antimatter you mean making neutral antimatter like antihydrogen that's much, much harder. The energies of antiprotons created from the nickel target are far higher than the ionisation energy of (anti)hydrogen, so the antiprotons need to be cooled and trapped then reacted with positrons to create antihydrogen.
The first significant amounts of antihydrogen to be made were created by the Alpha group at CERN. There's a nice video that describes their experiment here. They hold the antiprotons in a magnetic trap then adjust the field to allow them to come into contact with the positrons.
