To put it simply a line doubler increases the resolution of a video display by creating a sort of pseudo-progressive scan picture. To fully understand exactly what it is and does we'll need to understand the difference between progressive and interlaced scanning as well as how this is accomplished. First off, though, is the manner in which pictures are resolved on your screen.
The picture you're viewing on your video display is composed of two vertices: horizontal and vertical. Each of these is composed of a number of scanning lines, 525 horizontal lines for the vertical resolution and between about 240 (VHS) and 480(DVD) vertical lines of horizontal resolution. The vertical resolution is set by a standard (525 is the NTSC standard used in the US, Europe uses the PAL standard of 625 lines) and does not change, while the horizontal changes with the source. For our purposes we will be concerned with the vertical resolution.
While vertical resolution is a set standard it is not shown in one picture all at once. Instead most displays show it in interlaced mode. That is, in the first 1/60th of a second it scans the odd numbered lines (1,3,5,7...) and shows them. In the second 1/60th it scans and shows the even lines (2,4,6,8...). Since your brain interperets 60 images a second as a continuous image you don't see any flicker. Thus a normal image is shown in a frame composed of two fields at 1/30th of a second per frame.
Interlaced vs. Progressive
While interlaced (most TVs are interlaced) scan modes work in this manner this is not the only way. Progressive scan, rather than separating the image into even and odd lines simply shows the entire scan straight through progressively from line 1 to 525. This generally yields a much smoother picture with sharper edges. DVD players and video game systems have increasingly adopted progressive scan output, but most non-HD broadcast television is typically interlaced. To produce a progressive image from these you'll need to first de-interlace and then double the lines. True progressive scan is largely the domain of HDTVs these days but there are two main methods of viewing this de-interlaced, semi-progressive scan.
Improved Definition Television (IDTV) seeks to resolve this problem by storing the initial field in memory, but not showing it. It then adds in the second field and shows them together. This reduces the effect of distinct scanning lines, however it requires very fast memory leading to motion artifacts appearing when the camera is panned quickly. As well the image tends not to be as sharp.
Finally we come to the subject of this write-up. A line doubler is a very expensive device to provide a non-interlaced image without the problems of IDTV. Primarily produced by Faroudja they use a computer to estimate the information between the information in the original field. For example in between lines 1 and 3 there would be a new line 2 created by the doubler. In order to make this work its best though you'll likely need a high-end data-grade front-projection system with RGB and Synch inputs capable of scanning at 31.5kHz (double the scanning frequency of NTSC). Considering that the data-grade projector is likely to run many thousands of dollars alone before the thousands the doubler itself costs this is not likely to be something you'll be able to pick up at the local Best Buy. Nevertheless it does not discourage the more extreme videophiles and home theater enthusiasts.
Increasingly, however, video scalers are being used in place of line doublers. The primary difference being that video scalers are more "intelligent" and are capable of modifying the resolution, aspect ratio, refresh rate, and other features such as implementing 3:2 pulldown (a necessary process when converting 24 fps film to 60 fps video and commonly a part of the telecine process). In the modern era this is what you are most likely to find in your brand new HDTV or upconverting DVD player.