A telescope is primarily a light bucket - it collects light. The bigger the telescope, the more light it collects.  The light gathering power is proportional to the diameter of the telescope squared.  A 10-meter telescope collects 100 times more light than a 1-m telescope.
Light Gathering Power a D²
A telescope with a large mirror not only collects more light than a small telescope, but it also has a finer resolution, meaning that it has the ability to detect smaller details. The resolution is proportional to the wavelength of the light divided by the diameter of the telescope. If you increase D you get greater clarity in your images and finer resolution. You also get finer resolution by using smaller wavelengths. The magnification in a telescope only has to do with the eyepiece; it essentially magnifies the focal plane image and more detail will be seen only if there is finer resolution too.
Resolution ~ l / D

All telescopes collect and focus light, either with a  lens or a mirror.  A refracting telescope uses a lens to focus parallel light rays coming from the astronomical object to a common focus point behind the lens. In a reflecting telescope with a parabolic mirror, the parallel light rays are reflected off the mirror and they come to a focus point in front of the mirror.
 There are 3 reasons why mirrors are better than lenses:
1)  lenses are more  expensive and harder to make
2)  lens have to be supported at the sides only which causes the massive lens to sag under its own weight, causing distortion in the images
3)  light rays of different wavelengths are refracted by lenses a different amount so that the focal point is at a different place for each wavelength.  This is called chromatic abberation, which does not happen with mirrors.

The world's largest optical telescope is the 10-m (~30 ft) diameter Keck telescope, located in Hawaii on an extinct volcano at a very high elevation, where there is less atmosphere for the incoming light to pass through. Turbulence in the atmosphere causes blurriness in the images and limits the amount of detail you can achieve from the ground. In order to get higher resolution than you can get looking through the earth's atmosphere you have to put a telescope out in space, above the atmosphere. The best images are currently made by the Hubble Space Telescope.  The mirror in the Hubble is actually only 2.5-m and has much less light gathering power (roughly 11 times less) than the Keck, but because it is above the earth's atmosphere there is no blurring of the images, so the images are much clearer. The new technique of "adaptive optics" is changing that situation. Computers are hooked to a "deformable mirror" near the focus of the telescope. A "wavefront sensor" uses a bright star or a laser spot (high in the atmosphere) to measure the distortions caused by the atmosphere. The deformable mirror is then adjusted to remove these distortions. This has to be done many times per second. The field of view that can be corrected is rather small, so space telescopes still have their place.

Another reason to put telescopes in space is that at certain wavelengths, our atmosphere absorbs incoming light rays, so that some kind of light never reach the ground. There are satellites in space and balloons in the upper atmosphere with telescopes on them that observe gamma-rays, x-rays, and UV and infrared light. Much of the electromagnetic spectrum emitted by stars, planets, and compact objects can only be viewed from space, so this will remain a major asset of modern astrophysics. NASA has a "Great Observatories" program which consists of the Hubble, a gamma ray satellite, an X-ray satellite, and a soon to be launched infrared satellite. The "Next Generation Space Telescope" (8 meter class) will also be an infrared telescope. The European Space Agency also has several major observatories.

 Radio telescopes have to be much larger than optical telescopes in order to achieve the same kind of resolution, since resolution scales with wavelength. This technique is called "interferometry". The world's most powerful radio telescope is the Very Large Array, or VLA, which is made up of 27 25-m antennas arranged in a Y-shape in the desert of New Mexico.  Astronomers can use the many little antennae to simulate pieces of one large telescope and get the resolution that you would get with a telescope that had a diameter of 22 miles. Yet another technique ("very long baseline interferometry) links radio telescopes around the world to form the equivalent of an Earth-sized telescope! Of course, it has much less light gathering power than the full large telescope would (and is MUCH easier to build). There is now an effort to bring the technique of interferometry to the infrared and eventually the optical (it gets harder as the wavelength gets shorter). The 2 Keck telescopes, and the 4 8-m VLT telescopes will each eventually be linked together to form the equivalent resolution of ~100-m telescopes.

To take a spectrum of the light coming from an object, the light is focussed onto a narrow slit (to produce a narrow image. This image is passed through a prism or diffraction grating to spread the light out into a rainbow (each wavelength gets bent by a different angle).  You then take a picture of the rainbow with a camera and measure the intensity of the light at all different wavelengths, which is a spectrum. If there are wavelengths which have less intensity (because atoms absorbed them), the slit image will be dark there, producing dark lines in the spectrum (this is why we call them "spectral lines").