Hackerspace origins and evolution
The hackerspaces movement as we know it today largely originated in Germany. Soon after, an explosion of interest in hackerspaces resulted in the creation of new spaces all over the world, especially in Europe and North America. Mitch Altman has said there are now more than 1,100 hackerspaces in existence around the world, up from 20 or so in 2007. Hackerspaces.org maintains a list that attempts to catalog all of the world's hackerspaces.
The goals of hackerspaces are varied, but there is a common element of emphasizing education and invention. In the United States, most hackerspaces are 501(c)3 nonprofit educational organizations, and many hackerspaces in other parts of the world hold equivalent status. Hackerspaces promote education by hosting events at the Space, usually open to the public and free of charge, which range from soldering workshops to industrial sewing workshops to sessions on DIY residential electrical wiring to classes on cryptography, data security, and every topic in between.
Many hackerspace members also volunteer to take their skills on the road by participating in events at local schools, libraries, and other nonprofit institutions. A popular activity at some hackerspaces is the appliance repair café, in which members of the surrounding community are invited to bring in dead appliances (alarm clocks, toasters, microwaves, you name it), and members of the space will help them repair the device and restore it to a working state (or establish that it is truly beyond repair).
Hackerspaces and their members like to "dream big" and create BHAGs (big hairy audacious goals) for themselves. There is currently an initiative called the International Hackerspace Space Program that has a goal of putting a hacker on the moon by year 2023! Talk about audacious. As Mitch Altman points out, it doesn't matter if the goal is actually attained, as the process of pursuing the goal is its own end; the learning, the exploring and the creativity harnessed and energized by trying to put a hacker on the moon will be its own reward. Note that this is not an entirely "pie in the sky" initiative yet; several hackerspaces are participating in a DARPA-funded project to do space research.
Given this diffusion of activity, how do you know when a hackerspace has succeeded? A hackerspace is a success when it helps people create things they could not have before. How does that happen? Either because another member has the right skills to help push through a sticking point or because there are enough people doing awesome things that something that looked impossible becomes much easier. Another way of establishing success for a hackerspace would be when they, as Mitch Altman says, "take something that was traditionally very expensive and make it cheap." The explosion of availability of inexpensive 3D printers is a prime example of this kind of hackerspace success story.
What hackerspaces mean to you
Why should you care about your local hackerspace? Well, as should be obvious, they are centers of learning. They offer both community -- the kind of collaborative energy usually associated with the university experience -- and an outlet for creative expression. As an individual, if you are interested in broadening your personal horizons, learning new skills, finding outlets for your creative desires, or hanging out with interesting people, there are few better avenues than interacting with your local hackerspace.