Simply put, resolution is how clear and sharp are the pictures.
Usually you will see resolution measured in DPI or dots per inch or in the number of pixels in a picture.
If you see it measured in DPI what it is telling you is how many dots are there in one inch of a line in the picture. The DPI will vary according to where the picture is being viewed. Most of the time when pictures are viewed on screens, they are viewed at 72 DPI as the screens often are limited to teat resolution. However, you can get high quality monitors that can view at much higher resolutions, however for sending across the Internet to view on a screen people often use a resolution of 72 DPI.
Printers can have much higher resolutions with a default value often of 240 DPI. This means it has much smaller and close together dots than the 72 DPI, so the sharpness and clarity is much better. Scanners can be much higher although many people scan at 300 or 72 DPI depending on what the image will be used for. The higher the DPI or resolution means the more space it takes to store the image.
You will also see resolution expressed in the number of pixels in a picture. Pixels are basically the same thing as dots so easiest to think of them as that but there is a slight technical difference. I just got a new camera and it told me it has 18 megapixels images. What this means is the images will have 18 million dots or pixels at 240 dpi resolution. That makes the pictures very clear even when printed poster or bigger sizes. Personally, when cameras got to 5-6 megapixels that was all I needed. The pictures I take are not normally printed any larger than 8X10 inches and 6 megapixels is very clear. The additional megapixels really offers me no real benefit but it does sound neat to say my pictures have 18 million pixels or dots.
I got the new camera because autofocus quit last year on my camera (Canon Rebel SLR) after taking over 100,000 pictures but not figured exactly.
You want high resolution on pictures you will print and will also want a higher resolution printer. However, if the pictures are for viewing on the Internet using a resolution of 72 will make them load quicker. In Photoshop or what ever editing program you use you can change the resolution. It does not work to increase resolution; you just get graininess. However, if you reduce the resolution or do editing, I would suggest save as a new name so you still have original you can go back to in future to do stuff.