here's an attempt...
dpi is what you get from inkjet printers when you print a picture on paper of some kind. 300 dpi is pretty fuzzy for a photo, 600 is OK depending on the original of course. It refers to how fine the print head nozzles are I think.
ppi is pixels per inch which is the real stuff of photos. A picture of 72 ppi looks ok on most monitors but will look fuzzy printed out on paper, regardless of whether the inkjet printer you are using is printing at 300 or 1200 dpi, as the information just isn't there.
A picture of 250 ppi looks pretty good printed on most good quality papers. That is of course providing it is printed at original size with 250 ppi resolution. If you take a pic of 250 ppi which is say 5" x 2", that's 10 sq inches, multiply that by 250 ppi in length and by 250 ppi in height and you have 625 000 pixels, or a 625k picture, crudely put.
You can enlarge that picture to twice that size-20 sq inches, say 8 x 2.5 inches- and as long as you halve the resolution to 125 ppi, you still have the same original picture. The computer hasn't had to invent any pixels to fill any gaps. You've just stretched it, like printing it on a balloon and blowing the balloon up a bit more.
If you want it at double size - 20 sq inches- and want to maintain the 250 ppi resolution, the computer only has 125 ppi that it knows of at that size, so it fills in the gaps by averaging the colour of adjacent pixels. You have some control of this process by selecting bicubic interpolation, or other kinds of interpolation. However, you're getting it to create something from nothing, so it won't be as good as the original was.
If you knew you wanted it to be twice the size, you ought to have scanned it in at a finer scanner resolution in the first place, so that after enlarging it, the quality is maintained, and nothing is invented.
Scanners are a different thing. Most have a setting in dpi that you choose to have them scan at. However some actually mean ppi when they state dpi. Others say dpi as that is what they expect you to use for your inkjet output setting. I can't enlighten you there. You'll just have to figure out your own machine.
lpi is lines per inch, also called the frequency (of lines on a white screen originally used to create bromides for reducing a photo to half tones for say newsprint or magazine output). A laser printer can only print black, not grey. A photo to look realistic needs shades, so half tone pictures are the solution to get black to appear grey, by altering the amount of black on each dot. These dots are set a certain distance apart - 80 lpi frequency means 80 laser printer dots per inch. You can have a tiny pin prick of black, or none at all, or a big blob in each 'eightieth of an inch' of picture.
On laser printers with Post Script drivers, you can alter the frequency (lpi) down to 45 (crude but OK), 60, 80, 120 or more. 120 is awfully fine and not much use on plain paper- you won't see the difference much past 100 on standard paper. 80 is pretty ok.
You can also set the blob shape - square, diamond, elliptical or circular- for various reasons I won't explain here. You can also set the angle of the lines of the screen- vertical (90 degrees), 45 deg, or any other angle of choice.
There are reasons for altering this, principly to avoid moire, which is an interference pattern appearing and dominating the picture. With colour printing on laser printers, or screen printing, you set each colour at a different angle, to prevent this problem. You can do a search for more info on that.
So.... does that help?
Why are there these differences? Because of the different output 'machines' & technologies. What do you have, & what do you want to print it onto (cloth a coarse t-shirt or fine silk, mugs, glass, vinyl, standard paper, photo paper, newsprint, magazine, brochure...) and how big is it & how big is it going to be, and what equipment will be used for the intervening processes (which has its own limitations of quality & puddling), and how much is the customer's budget & how fussy are they & you about the quality of the end product... and what software do you have... etc.
There are lots of controls aimed at helping the pilot get a good landing on whatever's ahead!
That relates to bitmaps, or photographic stuff mostly, which is how things get printed. Vectors are mathematical, and help you create huge physical pictures while keeping the working file size small. They can be scaled to all sizes without loss of clarity because they are a mathematical formula for an outline, plus a fill inside that boundary. Bitmaps are individually memorised pixels which are independent.
A plotter plots or cuts outlines, but any photographic picture to be printed needs to be converted to pixels & dots, even if it was a vector pic to start with..
Clear as mud?