As a simple way to visualize resolution, the higher the resolution is, the smaller the pixels will be once printed. That's why you see a different size on your screen depending on the resolution when looking at the image's dimension; if the resolution is 300ppi for example, the pixels will be smaller and more concentrated. If the image is 72ppi, the pixels will be bigger once printed. It's better to not use the ruler on your display to measure the size of an image for this reason, but it's possible.
It's easier to simply trust the measurements and dimension of the file when looking at the "image size" in Photoshop (eg. inch, centimeters, etc.) That measurement is also the one that will be used in your other publishing software if you import your image. An image imported in InDesign with 1000x1000 pixels at 300ppi will look smaller than the one imported at 1000x1000 pixels at 72ppi for the reasons explained above.
In general, in printing, you can think of resolution as dots of ink; the smaller the dots, the closer they'll be and the less they'll be visible on paper. And the smaller these dots are, the clearer the image will be. That's why the requirements for printing are higher than for web use.
But it's not entirely true that 1 pixel once converted to be used for print will be shown as 1 dot. It depends on how the rip system will encode these pixels to fit the printer's quality. For example, if you use an image at 30ppi (30 pixels-per-inch) and print it, there will more than one dot to reproduce that one pixel as seen on the screen and the printed image will look blurry. If the printer is a high resolution one (eg. uses 300dpi), it will always fill with extra dots the missing "pixels" on its grid.
As the unit for the resolution says, if you have 30 pixels-per-inch (30 ppi), they'll logically be "bigger" than a 300 pixel-per-inch (300ppi) image if printed. The printer will not create one bigger dot for each pixel, it will split that big pixel into many small dots instead (see image above) and fit as many it can according to how many line-per-inch it can print in that grid. In offset printing, the size of the dots will only change depending on the color density of each color separation (Cyan, magenta, yellow, black and spot colors) and the quality of the printer. In short, the quantity of dots represents the quality of printing and the real size represents the color density.
Some precision: Pixels are used as a unit for screen display (eg. web projects), ideally not for print. Even if they have a square shape, their real "physical" length will change depending on the device you use and the aspect ratio (eg. web vs video.) That's why when referring to size/dimension for print projects, it's more precise to simply use the resolution and the measurements using imperial, typographic or metric units of length (eg. centimeters, points, picas, inches, etc.) It's closer to the final expected printed result than pixels. More details on pixels here.
Digital printing looks better at 200dpi and up and the offset printing should be at least 266dpi (preferably 300dpi and more for color, and 600dpi for black and white texts). If you print on a laser printer in your office, you can go as low as 150dpi.
By the way, ppi or dpi are both terms used for resolution but they represent the same value in software like Photoshop (see this post for more details: Why does Photoshop call ppi "resolution"?)
How to change resolution on a low resolution image in Adobe Photoshop
Regarding the question about your low resolution images... you can still salvage them within some limits or at least try.
As I explained above, the higher the resolution, the smaller the pixels. What you can do to salvage your low resolution files is to raise the resolution of your image proportionally with your image dimensions.
I don't know what's the dimension of your images, but maybe you'll be surprised at the size they can be printed even if right now they're at 72-100ppi. If your image is really big at low resolution, it can be printed about 1/4 of its size.
Now, what you can do is to open your file in Photoshop, and go on the top menu "image" and then select "image size."
As you see on the screenshot below, you should uncheck the box "resample" before changing the resolution or the size of your image. Otherwise it will create "fake" pixels and increase the resolution of your image artificially which will make it look very low quality.
After you uncheck that "resampling" box, you should see the "link" that ties together the height/width/resolution. Change your resolution to 266ppi and see how the size changes. The new dimension is the minimum print size you can use your image for. As I mentioned, you can use 200ppi, and at the limit even 166ppi for digital printing (ex. Xerox, color Laser.)
Also, you will notice that the number of pixels didn't change at all beside it's the size of the pixel that was changed, not the quantity.
Now you can have a look at all your images and see how big they can be printed. If you're fine with the size, you can simply save your image with this new resolution.
If you really need to increase the size of your picture, there is some tolerance of about 20%; that means you can increase the size of your image of about 20% before seeing significant pixel distortion. To do this, you'll open your "image size" again, but this time check the "resampling" box, and change the size of the image to 20% bigger. This is to be done with precaution and if the result doesn't look good when you look at your image at 100% of its size on the screen, that means it's going to look probably as bad once printed!
Source: Pixel image - pn-design.co.uk, Resolution vs size image - i.stack.imgur.com, DPI image - e-education.psu.edu, LPI image - ajslabels.com
Edit: I added some precisions due to the confusion about measurements and units.