BooksTom Leinster, General topology.
These are some notes for a course that Tom Leinster recently taught at the University of Edinburgh, and has made freely available for download. They may not be proofread as carefully as a published textbook. They begin with the metric space viewpoint, and move quickly to the definition of topological space (somewhat more quickly than Sieradski, below).
Allan Sieradski, Introduction to topology and homotopy.
The general approach to the book is not so different from that of Munkres. It has more pictures in it, and spends more time on metric spaces. (In particular, it's a good place to read about metric topology and the motivations for the definition of a topological space.)
Currently (as of January 2015), a Google search for "Sieradski topology" comes up with a pdf file hosted on a researcher's homepage. I am unable to determine whether this copy is legitimate, so will not link to it; but I'm also unable to determine that it is not legitimate, so I here mention it. It appears to be out-of-print, which is a shame, as it is a nice treatment. (Used copies are available on Amazon, but some are rather expensive.)
Mark Armstrong, Basic topology.
Armstrong's book takes a somewhat different approach, beginning with a stronger motivation from surfaces and Euler characteristic. It also has somewhat different topics in it -- it looks to have nice treatments for example of triangulations and homology.
John Conway, A course in point set topology.
Conway's book also begins with metric spaces. It has somewhat fewer topics than the books listed above, which might be at times more approachable.
WebsitesWikipedia is, perhaps surprisingly, not a bad source for background material on basic definitions and results in many areas of mathematics, including topology.