What is reconstructionism?

Short Answer:A movement within a number of religions to reclaim the religious traditions of the past as accurately as possible.
Long Answer: Reconstructionists practice a large number of different faiths based on indigenous religions. While few reconstructionists embrace the umbrella term of Pagan, prefering words like "Heathen" or simply the proper name for a practitioner of their faith, "Pagan reconstructionists" often use the term to distinguish themselves from similar movements within Christianity and Judaism.
Some good, specific Reconstructionist links:
Hellenion Greek (Hellenic) Reconstructionism
Imbas Celtic Reconstructionism
Religio Romana Roman Reconstructionism
Ásatrú "Norse" Reconstructionism
The above are good examples of the top four "Pagan" reconstructionist movements. These movements are NOT Wiccan, and Wicca is not Reconstructionist, just in case you were confused.
More Info:
"What do you mean 'Indigenous belief'?"
-Indigenous beliefs are remnants or parts of indigenous (natural to or original to an area) religions.
"Is there a difference between Paganism and Neo-Paganism?"
-In general, no. Many people use the term "Neo-Pagan" to distinguish between Ancient Indigenous Religions, which they call "Pagan" and Modern religions, which they call "Neo-Pagan." For most of these ancient religions, Pagan is a perjorative, so it is best to call self-identified Pagans Pagans, and others by the proper term for their religion, or at least the generic term (like "indigenous religion of Ancient Sparta").
Citations: Beliefnet.com, Witchvox.com Go to next Question