Fortunately for us, the earth's atmosphere blocks X-rays. Unfortunately
for astronomers, this means we need to be above the atmosphere, in a
balloon, rocket or satellite, to observe extra-solar X-rays.
How does one built an X-ray telescope? X-rays do not reflect off mirrors the same way that visible light does. Because of their high energy, X-ray photons penetrate into the mirror in much the same way that bullets slam into a wall. Likewise, just as bullets ricochet when they hit a wall at a grazing angle, so too will X-rays ricochet off mirrors. These properties mean that X-ray telescopes must be very different from optical telescopes. Basically, they contain many almost parallel plates, that gently nudge the incoming X-rays into the new direction you want them to go in. See http://chandra.harvard.edu/xray_astro/ for a nice diagram and more detailed info.
The sun (we're really close!) was the first source to be detected in the X-rays, in 1949. The technology improved in great leaps, and it came somewhat as a surprise to detect X-rays from Es that were not associated with point sources. Why am I talking about point sources? Recall from the stellar part of the course the concept of X-ray binary. In these binary stars, hot gas associated with mass transfer radiates in the X-ray band. Such (binary) stars are also seen in Es. But subtracting these sources from the X-ray image, we still detect extended X-ray emission from the galaxy. The process that produces them is called thermal bremsstrahlung.