In the spring of 1961, Explorer-11 was launched. This satellite observed the first gamma-rays from cosmic sources. Unfortunately a detector failure in the fall of 1961 cut the mission short, but while it was up, Explorer-11 detected solar flares, radiation from the Van Allen belts and 22 gamma-ray events from random parts of the sky.
In early 1962, Ranger 3 was launched toward the moon - the first successful launch of a satellite to the moon. The satellite crashed into the moon (as was planned), but its gamma-ray detectors made the first detection of a diffuse gamma-ray background (this background is believed to be unresolved blazars).
In the summer of 1962, an X-ray detector on a sounding rocket was flown. The goal of this flight was originally to detect X-ray fluorescence of the moon due to the Sun's wind; it failed to detect such emission from the moon. However, the detector serendipitously discovered extra solar X-rays from a source named Scorpius X-1 (the first X-ray source seen in the constellation Scorpio). In addition, it detected a diffuse X-ray background (some of this has been resolved into sources and the rest is believed to be from unresolved active galaxies).
In the late 1960s, the Vela satellites were launched to monitor the Earth for gamma radiation. A strong signal of gamma-rays from the Earth would have resulted from nuclear weapons testing, so these satellites were the U.S.'s way of monitoring whether or not the Russians were complying with the Limited Test Ban Treaty of 1963.
Instead of detecting gamma-rays from the Earth, these satellites discovered large, short bursts of gamma-rays from cosmic sources. These mysterious objects were not reported to the rest of the world until the early 1970s (as a protection of U.S. security). In fact, the nature of gamma-ray bursts remains largely a mystery even today (see Gamma-ray Bursts in the Sources section of this tutorial).
Such discoveries spawned several generations of X-ray and gamma-ray satellites. Uhuru (a.k.a. SAS-1) was the first satellite dedicated solely to X-ray astronomy and was launched in December 1970. A series of three High Energy Astrophysics Observatories were launched between 1977 and 1979 (HEAO 2 was renamed the Einstein Observatory after its successful launch).
More recently, two of NASA's "Great Observatories" have been dedicated to high energy astrophysics. The Compton Gamma-Ray Observatory (CGRO) was launched in 1991 and operated successfully for 9 years (it was de-orbited in June of 2000). The Chandra observatory was launched in 1999 and has been taken excellent X-ray data for the past two years.
For a more complete history of high energy astronomy, check out NASA's High Energy Astrophysics Science Archive Research Center's A Brief History of High-Energy Astrophysics.