> I found out that my printer has a resolution of 600 dpi (dots per square inch?).

That's usual. Higher resolution are too expensive for the home user; an Agfa 2400-dpi thermal transfer printer can cost up to USD 20,000.
At any rate, dpi is "dot per (linear)inch"; this means that if you print a one-inch segment, it contains 600 "dots".

> How does this compare to pixels per square inch?

They are always linear inches. A 1x1in. square will contain 600x600 dots.

Each output device has its resolution in dpi. A good PC monitor is about 96 dpi. This means that your monitor will light up 96 dots to display a 1 inch segment (1 dot=1 pixel), but your printer will print 600 dots (1 dot = 1/600in. = 96/600 pixels) to print the same segment.
Of course, scanners, printers and monitors always have a resolution expressed in dpi; monitors also work in pixels, where 1 pixel = 1 dot on the monitor. Generally, the dot is the smallest unit that each device can process; a pixel, by definition, is the smallest unit that can be used to display an image (so we have the 1 dot = 1 pixel relation on a display).
Dots are device-dependent: a dot on the screen is 1/96 inch, on the printer is 1/600 inch and so on.

By the way, "dots" and "points" aren't the same: a point is 1/72.2 inch (in typography) or 1/72 inch (on a PostScript device) - points are device independent. Font size are usually expressed in "points", so a 8-points Arial text will be always the same height on a 96dpi screen or on a 300dpi printer.

Regards,
AP.