Resolution is the number of pixels per linear length of an image. The most widely used units of resolution are ppi (pixels per inch) and dpi (dots per inch). For an example, each inch of a 120 ppi image would contain 120 pixels horizontally as well as vertically. Following image is an example of 24 ppi.

Different medium uses different resolution units. For examples, digital images for web and display monitor screen are measured by ppi (pixels per inch) or dpi (dots per inch). In the printing industry, the standard is lpi (lines per inch), which is the number of halftone dots per linear inch.

Different resolutions generate different qualities. Following example shows the same image in different resolutions.

As you can see here, image with the highest resolution generates the most detailed and clear image, but at a much larger file size. And the lowest resolution yields the worst quality at the smallest file size. So it is very important to balance the quality/file-size ratio. Typically, for image publishing, quality is overwhelmingly more important, and for web graphics, file size takes more consideration.

So what is a good resolution for your work? Depends : If it's a black and white image (1-bit), then there's absolutely no reason to have the image resolution different than the print resolution. If the print/display resolution is 300 ppi, then the file should be 300 ppi as well : If the file has more than 1-bit colors, then generally, they use resolution that is anywhere from 150% to 200 % of the final output. If the image resolution is smaller than 100% of the final output, the details are lost. If the image resolution is higher than 200% of the final output, then the file size of the image become too large without much of visible benefits.
In the printing industry (US standard), for typical professional magazine prints, 133 lpi is the standard, so 267 ppi for image resolution used to be the most popular choice (about 2 x 133 lpi) among professionals. For high quality artbook prints, the print standard is 177 lpi, and therefore 354 ppi is the most popular resolution. However, with the advancements in modern printing technology, there are many more different resolution standards (such as 300 and 600 dpi) which works just as well.

For the web publishing purpose, traditionally 72 dpi for old Macintosh, and 96 dpi for PC were used to determine the size of display. However those numbers lost their merit more or less because of the advancements in display technology, and now those resolutions are taken more or less as a general guideline rather than the absolute value. :

One term that comes up again and again in digital image is anti-aliasing. What does it mean?
Well, aliasing mean a staircase-like effect (jagged edges etc.) in a bitmap image. Solution to this artifact is anti-aliasing which averages the brightness values of each pixels with its neighboring pixels, resulting smoother image.

OK :) This is a good place to stop for today :)