Printer DPI and PPI Ratings, General
Dots per inch stands for the maximum number of tiny spots of ink that the printer can place in a straight line where the spots are theoretically small enough (i.e. ignoring spreading or smearing effects of ink on paper) that if placed in every other such dot position leaving white space between them, the spots can be individually distinguished.
Pixels per inch stands for the maximum number of unique positions in a straight line that the printer can place an ink spot under control from the outside world, namely from a computer connected to the printer.
Lines per inch stands for how close thin parallel lines can be printed and still be distinguished in the finished printout. The spaces between the lines count as "lines".
Pixels per inch and dots per inch originally referred to the same thing. The printer mechanism was under the direct control of the computer and was physically positioned and placed dots as directed by the computer. Back then, most printer mechanisms were limited to placing dots only in positions suggested by a grid of dots X per inch horizontally and Y per inch vertically, for example 100x100 dpi
Nowadays, many printers put dots "wherever they want" as opposed to in positions suggestive of a horizontal/vertical grid. Still there is a minimum dot size and a minimum dot spacing.
A picture file (image file) represents pixels in a uniform horizontal/vertical grid pattern. And the printer needs to make a finished picture of the size, say 5x7 inches, that the user chose regardless of the number of pixels in the picture file. To simplify the process of relating the pixel count in the picture file to the possibly non-uniformly spaced dots on the paper, the printer or its supporting software may generate a temporary intermediate picture file with a set number of pixels per inch. The printer may have, internally, several choices of ratio of pixels to dots and the published rating can be the largest ratio except that the published rating may not exceed the dpi rating. Therefore there might be three "per inch" values involved at a given time, the pixels of the original picture file, the pixels per inch that the printer works with, and the dots per inch of the printer mechanism.
Pixels per inch is usually not mentioned with printers. All printers come with their own software (including parts called drivers) to install on your computer. Usually the software does not let you exercise control over individual dots using your picture file. Rather the printer takes your picture file or data file and uses its own built in logic to lay down the dots and create the printed output. We are led to believe that a printer's ppi is usually a fraction such as a half or a third of its dpi rating.
When a temporary picture file is created, there are at least two levels of software in use. High level software (which may run in your computer) takes your picture file and creates the temporary file. Low level software runs in the printer, takes the temporary file and controls the dot size and dot placement on the paper.
Sometimes a printer is advertised using a phrase such as "300 dpi 1200 dpi quality". This means that the printer has some way of making dark edges on a light background appear smoother than the first number would otherwise suggest. A printer with 300 dpi 1200 dpi quality definitely cannot resolve alternating dark and light pixels less than 1/300'th inch each. But curved and diagonal lines and color boundaries should not have jagged edges suggesting individual dots rigidly positioned on a grid with a 1/300'th inch pitch.