An introduction to fractal theory, also with nice pseudocode (useful if you're a CS person and want to write simple programs to simulate this sort of thing). Explanations of how complex numbers are involved in the Mandelbrot/Julia sets.
http://www.cygnus-software.com/theory/theory.htm

Another intro to higher-dimensional fractals, with code. Website has JavaQuat, which you can use to visualize quaternion-based (and other) 3- and 4-D fractals.
http://www.lystad.us/fractals/docfiles/higher-dimensions.html#SPACES

An excerpt from one of Mandelbrot's articles on fractals and nature. A good introduction, though parts will go over your head if you haven't read up on chaos theory.
http://www.fortunecity.com/emachines/e11/86/mandel.html

A low-math intro to fractals. Most interesting for the addendum, which is a neat trick you can do in Excel to graph iterations of the Mandelbrot formula (or others). http://members.lycos.nl/ton_koppens/Tutorial01/test.html


Dimensionality:

(General/1D/2D)
Fractal Dimensionality (some info also on Peano and Koch curves) http://www.bath.ac.uk/~ma0cmj/FractalDimensions.html

Cantor's Dust (0D/1D)
http://ems.gphys.unc.edu/nonlinear/fractals/defs.html
See it! (at least the first 5 iterations of it...) http://www.ecu.edu/si/cd/interactivate/activities/cantor/index.html

(1D/2D)
Von Koch curves (Koch snowflake and related curve series) http://perso.wanadoo.fr/charles.vassallo/en/art/von_koch.html

(1D/2D)
Images of *many* 1D/2D line-generated (more precisely, L-system, or Lindenmayer System) fractals, including Cantor's Dust, the Sierpinsky Gadget, and more.  Also yet another program you can use to generate them.
http://www.cs.pdx.edu/~coscorrr/programs/fractals/images/lsystems/

(2D)
Markus-Lyapunov fractals
(another set of 2D fractals, similar to Mandelbrot but with a different formula) http://perso.wanadoo.fr/charles.vassallo/en/lyap_art/lyapdoc.html

(2D/3D)
2D and 3D triangle fractals. Also some neat building blocks he designed, and connections with Escher's drawings.
http://www.tabletoptelephone.com/~hopspage/Tetfract.html

(3D)
Fractals applied to sphere packing problems in 3D. Pretty cool, and a nice example of a 3D space-filling fractal.
http://www.cg.inf.ethz.ch/~peikert/papers/apollonian.pdf

(3D)
3D landscape generation from multidimensional Cantor's Dust variants. If you reduce the iterations all the way (ctrl-click the windows repeatedly), you can see the underlying pattern. Alt-click to go back upwards and see how it evolves.
http://www.people.nnov.ru/fractal/3Dapp/Cantor.htm

(3D/4D)
A fairly clear explanation of how the Mandelbrot/Julia sets are 4D (also documentation/source for Gnofract4D, which, conveniently, you can use to visualize them) http://gnofract4d.sourceforge.net/manual/maths.html

(3D/4D)
Images of some 3D fractals (and 3D cross-sections of the 4D Juliablot set). Not very large images, though. The 3D computer-generated sculpture is pretty interesting. http://emsh.calarts.edu/~mathart/portfolio/SPD_Frac_portfolio.html


Other:

Lego fractals.
http://www.sci.usq.edu.au/staff/robertsa/legofracs.html