The shorthand format for resolution in screen or camera taking - a number followed by p, or a number followed by i, means p - the "progressive" digital film format or i - the "interlaced" digital film format.
The number that goes before i or p simply means one of the two lines of dots that make up the resolution - it is always the vertical or side line given. And that is going to be 1080 dots (pixels) exactly with 1080p.
What this shorthand resolution format doesn't tell you is the number of dots in the horizontal line of pixels. It can be any number.
1920 is the most common value for the number of horizontal pixels these days in 1080p resolutions - it is the standard "Full H.D." format, found in Blu-Ray discs and also produced now for most 1080p home T.V. sets. That resolution is 1920 x 1080. But that is not a definition of what 1080p is, though 1080p can be and is that. 1080p is also 1440 x 1080 pixels in progressive digital film formation.
Progressive means each frame of the film taken (each still photo it is in progressive format) is a full frame of the full number of resolution pixels - i.e. 1920 x 1080 films show every single frame made up differently, separately with that number of pixels, and the frames just progress in sequence, with nothing to smoothen them nor any pixels cut out in order that less memory is used. A film can be shot in any value of frames per second - whatever value they may be shot in, cinema films are usually edited to be released in 24 frames per second. So there are 24 full value frames of 1920 x 1080 pixels every second with that 1080p resolution. Typical home camcorder frames per second are 30 fps and 60 fps for rapid motion filming.
Interlaced is the other digital film format. Not all pixels are used in each frame. A frame is usually broken into fields, a number of fields make up a frame. Usually there are significantly less frames per second than with the progressive format, but as each frame is made up of a number of fields, this is also high quality, high definition digital film. Each field is not taken (or transmitted / played back if talking about T.V. sets which give i signals) with the full number of available pixels - different parts of the full frame will be taken in different frame, depending on motion and other variables (the machines work so quickly). The fields are played back rapidly, making each frame advance, to kind of give the effect of the whole resolution being used - as it is acutually (usually), but just not all at the one time. i film often looks more 'dancy' or slightly jerky to 'p' film.
Most of the world grew up for decades watching i - interlaced - film, which was also the most popular analogue format in standard definition, for beaming film to TV sets in homes. Now the world is slowly switching to transmitting tv in p - progressive format. (Some places have had progressive TV signals available to receive for many years, though). I think i is still most commonly found, if you look all around the world.
Perhaps I understand a little bit why some people are reluctant to associate the screen resolutions which they see TVs have with their laptops and laptops generally. But I guess that is only not being used to it, and also it's a big product of market programming - all those boxes with 1920 x 1080s and 1280 x 768s or whatever - it's almost natural to want to set those quite annoying things aside to "the world of the new TVs" which have come upon us. I guess it annoys people even with TVs - 12 or 13 years ago you'd have to search all day or longer to find an advert for a Television - the very idea of it was weird, now it seems to be more a part of life than work or hobbies or spending time with the children. And it's natural to want to sideline it only to those big boxes with the new TVs that we didn't have growing up.
But actually, those boxes contain more or less the same thing as a laptop screen in the modern day. Some digitals screens are set to TV use only, some to computer monitor use only, and some to both. But, essentially, they're virtually the same digital item - whether on a laptop or in one of those big HD TV boxes in a store. It's natural that you want to know the resolution of your laptop screen. Your laptop screen is made up of dots - that's all - and most people are aware by now, how many there are is usually the biggest factor in how good it is to look at. If you play DVDs on your laptop you'll want to know (though it's not so relevant for that, unless you have a 17 inch laptop and an DVD upscaling laptop, if there are any, I don't know). For your home movies shot on HD home camcorders which are inexpensive these days, then your laptop resolution really does matter.
The only thing is - your laptop is very unlikely indeed to be anything more than the standard resolution for the size of the screen, nearly all NEW (new = edit addition, sorry for any confusion) laptops are the same for each size of screen in inches (released within a year or two of each other, that is) - so yes, that's a reason why it's a bit annoying to see the figures for screen resolution for laptops all of the time. But, with a really budget end laptop, it could be a little less than the modern standard resolution for the size (in inches) of screen.