Recommended books:

The Right Stuff by Tom Wolfe
List Price: $10.88
Availability: also as e-book, but avoid this version
Paperback - 436 pages (1979)
Farrar, Straus, and Giroux; ISBN: 0-374-25032-4

The Heavens and the Earth : A Political History of the Space Age by Walter A. McDougall
List Price: $45.32
Availability: also as e-book
Paperback - 584 pages Reprint edition (September 1997)
Johns Hopkins Univ Pr; ISBN: 0801857481

Korolev : How One Man Masterminded the Soviet Drive to Beat America to the Moon by James Harford
List Price: $16.90
Availability: Amazon
Paperback - 432 pages (April 1999)
John Wiley & Sons; ISBN: 0471327212

Here is a diagram of the Earth's magnetosphere (solar wind is coming from the left):

Solar wind plasma which is captured onto the Earth's magnetic field lines is accelerated to high energies, and deposits this energy in the region where the magnetic field lines enter the Earth's atmosphere near the north and south magnetic poles.  This is called the auroral oval.

Close to the Earth's magnetic equator, and starting about 400 km above the Earth's surface, there are two belts of trapped high-energy plasma called the Van Allen belts. The inner Van Allen belt poses a danger to astronauts (and instruments) who are in spacecraft with orbits that pass through this belt.  The Van Allen belts were discovered by instruments on the first US spacecraft, Explorers 1 and 3, in 1958.

Sketch of the Van Allen belts:

The mastermind of the Soviet space program, the man who designed the Semyorka and who is generally given credit for Sputnik-1 and a subsequent series of spectacular Russian space achievements that startled the USA, was Sergei Korolev (pronounced sur-gay koralyov):

At the end of 1957, the US was trying desperately to catch up with the Russians.  Here is a description from an official NASA history.
(Orders of Magnitude: A History of the NACA and NASA, 1915-1990.
By Roger E. Bilstein, University of Houston, Clear Lake.
Published as NASA Special Publication-4406 in the NASA History Series, 1989)

"On 6 December 1957, the much-touted Vanguard test vehicle rose about 3 feet from the launch platform, shuddered, and collapsed in flames. Its tiny 3-pound payload broke away and lay at the edge of the inferno, beeping impotently."

The next year, 1958, the Russians orbited Sputnik-3, which carried scientific experiments, including a Geiger counter to measure high-energy particles in space.  However, the tape recorder failed and they couldn't get their data back properly.

 

That same year, the US launched Explorer-1 on a military Redstone (Jupiter) rocket.  This was the first US satellite, carrying James Van Allen's Geiger counter experiment that made the historic first space discovery.

Again, from the official NASA history:
"Clouds of gloom deepened into the new year. Then, finally, a small rift. On 31 January 1958, an American satellite at last went into orbit. Not Vanguard but the ABMA-JPL Explorer had redeemed American honor. True, the payload weighed only 2 pounds against the 1100 of Sputnik 2. But there was a scientific first; an experiment aboard the satellite reported mysterious saturation of its radiation counters at 594 miles altitude. Professor James A. Van Allen, the scientist who had built the experiment, thought this suggested the existence of a dense belt of radiation around the Earth at that altitude. American confidence perked up again on 17 March when Vanguard 1 joined Explorer 1 in orbit."
 
Diagram of Explorer 1, the first US satellite:

The Jupiter rocket was also an IRBM (intermediate range ballistic missile), and it figured in the famous 1962 Cuban Missile Crisis.

In 1973, the first signs of a thaw in the space race began to appear.  Here are pictures that I made in Moscow:

Official word came from the Soviet government -- there was to be "cooperation" with the Americans.

Two years later (1975), there was the Apollo-Soyuz test project, the predecessor to the International Space Station.

Now (2013), fifty-six years after the start of the Space Age, the USA is still cooperating with the Russians.  Here is a powerpoint file that describes how the great sci-fi author Clarke foresaw a cooperative mission to Jupiter.