Why is it difficult to study extreme physical conditions in the lab?

Until the late 20th century, scientists were not able to study extreme physical conditions — such as those in stars or during the first moments after the Big Bang — in the laboratory.

The temperature and pressure inside stars are extremely high. This means that particles such as protons, neutrons, and electrons react with higher energy than is normally found in the laboratory. In the early universe, matter had so much energy that even protons and neutrons did not exist. Matter took the form of even more elementary particles, called quarks and gluons.

Thus the problem was to create highly energetic collisions between particles, to study the outcome of these collisions and understand the fundamental forces and properties of matter.

How the particle accelerator solves the problem.

The particle accelerator, one of the most complex machines ever built, is the device that solves this problem. The particles to be collided — protons, perhaps, or electrons, or more massive particles such as the nuclei of copper or gold atoms — have an electric charge. The accelerator is designed to take advantage of this charge to get the particles up to high speed before a collision.

Electric and magnetic fields line the tunnel through which the particles will pass. The electric and magnetic fields are designed to interact with the charge of the particles, guiding their paths and boosting their energy.

As particles pass through the accelerator — which may be in a long straight line, or a ring (as Fermilab's is) — the particles gain speed and energy. They may travel almost at the speed of light before finally colliding with other accelerated particles or with a stationary target.

Aerial photo of the Tevatron at Fermilab. Credit: Wikipedia

Sometimes the particles involved in the collision are antimatter particles. Antimatter particles are like "reversed" versions of normal matter: an anti-electron has the same mass as an electron, but a positive charge, and an antiproton has the same mass as a proton, but a negative charge.

Whatever the particles involved in the collision, the result is an array of new particles emerging from the collision and flying off in all directions. Electronic detectors record the passage of these particles, or measure their energy. Physicists then study these records to investigate the parameters of the collision, rather like accident investigators studying debris from a car crash.

There are more than 70 particle accelerators in the world.

Particle accelerators are found at major universities and national laboratories in the United States, and at a number of national or international laboratories in Europe, Asia, Africa, and Australia. See this list of particle accelerators around the world for details.
The Relativistic Heavy Ion Collider, RHIC (pronounced "rick"), at Brookhaven National Laboratory in Upton, New York, is famous for experiments that reproduce conditions in the early universe. By slamming the nuclei of gold particles into each other, physicists there have created a sort of "soup" of fundamental particles, called a quark-gluon plasma. Read this Discover Magazine story on RHIC reproducing Big Bang conditions.
At a facility at Florida State University in Tallahassee, Florida, experimenters are trying to recreate rare particles that are created in supernova explosions.