Halftoning means using different patterns of dots to simulate a greater number of color shades. LPI comes from the world of photography while DPI comes from the world of design. Whether it makes sense to speak of DPI resolution for a raster image depends on what you'll be using that image for. Most magazines, such as Time, are printed with 153 LPI or less. Halftoning masks are the patterns used to create the shades of color or levels of gray seen in the lines per inch on the printed media.
Most masks are square.
Let's say you have a printer which can do 300 DPI, that is, it can print 300 dots in an inch. That is the effective resolution of the device, since you are interested in nice shaded printouts and not in single bilevel dots. Of course, the larger the halftone size, the more shades you'll get, but the lower the effective resolution will be. If you are only going to display an image on your screen, then perhaps speaking of DPIs in the image is pointless. You'll be mapping one pixel in the image to one pixel on your display, so how physically big the image looks will depend only on the size of your monitor.