Producing unstable particles
Some elementary particles are stable, some eventually decay to other particles. The moment at which an unstable particle decays is random, but the average lifetime (and the statistical distribution of lifetimes) for any type of particle is well-defined. Typically these lifetimes are extremely short by human standards, tiny fractions of tiny fractions of a second. This means that if we want to study unstable particles, we usually have to produce them close to our detectors.
Producing particles is very easy in theory. If one accelerates particles and smashes them together at high collision energy, a part of this energy may be converted into the mass of new particles. This possibility is encoded in the world's most famous physics formula: , which basically says that mass is a form of energy. It also says how much energy a given mass corresponds to, with the conversion factor being the square of the speed of light. Mass energy can be converted from and to other forms of energy.
Producing particles is very hard in practice. At least the ones that we are currently most interested in studying. One reason is that many of them are heavy, so we need very powerful accelerators to get enough energy into the collision. These machines are challenging to build and expensive. Another reason is that many are only produced very rarely in a collision, so we need to collect a lot of data to analyse. This requires a lot of collisions happening in a short time, fast detectors, and massive data storage and computing capacities.