An emission line is light emitted by excited gas atoms or molecules at a specific wavelength determined by the atomic or molecular species doing the emitting.

When an atom or molecule is excited by some process, one or more electrons within it can be raised to higher "orbits" or energy levels. (In molecules, you can also excite vibrational or rotational excited states, with emission in the infrared and microwave parts of the spectrum respectively.) When raised to an excited state, the atom or molecule tries to drop back to a lower energy level, and does so by emitting a photon with the same energy as the energy difference between the upper and lower electronic state.

Emission lines are used in physics, chemistry, and astronomy to determine what kind of gas is doing the emission. Every element has a different electronic structure, and will thus have a different emission line fingerprint. For example, atomic hydrogen has emission lines in the optical part of the spectrum, with the most prominent being the line at 6563 angstroms corresponding to the (3-2) electronic transition. You can also use the relative strengths of various emission lines to determine the temperature and density of the gas. This is used in astronomy to take the temperature of star-forming regions. The red color of nebulae like the Orion nebula comes from hydrogen atoms, but we can see oxygen, nitrogen, and sulfur atoms giving off emission lines as well.

The principle of emission lines is also used in "neon lights," so named because someone once figured out that if you put neon gas in an evacuated glass tube and put an electrical current across it, you get a lovely red glow with a wavelength around (IIRC) 6300 angstroms. In this case, there is an electronic transition in the neon atoms with an energy equivalent to that of a 6300 angstrom photon. Different gases give off different colors, so you can pick the color you want your sign to be by filling it with different gases. Sodium (which is what is used in street lights) gives off a pale yellow-orange light for example. Monochromatic lasers also generate beams with specific wavelengths using this principle (for example, helium-neon lasers). You may have also studied this in chemistry class by putting a chemical solution on a wire loop and placing it in a Bunsen burner flame. The flame excites the atoms making them emit light, which changes the color of the flame briefly.

I don't know of the exact origin of the term "lines," unless it simply comes from the usual method of performing spectroscopy. When taking a spectrum of light, one usually passes this light through a linear slit before dispersing it into its constituent wavelengths with a prism or diffraction grating. When this is done, if the light is composed of emission lines, they will show up as images of the slit at specific points on your detector (i.e. as bright lines). If the light comes from a continuum source, the "line" will be smeared out into a solid band (a rainbow). This isn't always the case though. There is an imaging spectrograph on the Hubble Space Telescope that actually smears the whole image out over the chip, making an image of the object at each wavelength. Images in different emission lines show up at different points on the detector. They took an image of the SN 1987A supernova remnant this way. I guess then you'd call it an emission... blotch?