First off, Christian Reconstruction has nothing to do
with Christian Identity. While any particular Reconstructionist
might or might not be racist, there is no racism built into
their theology. Neo-Nazis and the Ku Klux Klan aren't
Identitiests, though they can form alliances with them, and
Identitiests might join the Klan; just because Christian Identity
is extremely racist doesn't mean that all racists are somehow
connected to Christian Identity. And while they might like to
overthrow the government, they don't necessarily want to start a
theocracy, and their aim in the overthrow surely isn't to bring
about the end of the world.
*Phew!* Now, onto what they are. Christian Identity
(I'll abbreviate this as CI) did start with the British
Israelites, who believed that the Anglo-Saxons/Aryans
were one of the Lost Tribes of Israel. They based this strange
belief on the fact that some of the Anglo-Saxon words that make up
English are similar to some Hebrew words. Of course, if they
knew any linguistics, they'd know that this sort of thing just
happens by chance, but they didn't, so they looked on this as
incontrovertible proof. CI took this one step further, claiming
that Aryans were the true Israelites, not the people
claiming to be Jews. This is one thing that all CI sects have
in common, and is where it got it's name from: Aryan Christians
realizing their "true" identity.
As for the question "Who are the people who claimed to be
descendants of the Biblical Jews?", this is the primary theory:
when Moses lead the Aryans out of Egypt, a rag-tag group of
degenerates followed them, and it was these folk who caused
most of troubles for the Chosen People (the Golden Calf and
such). Although I haven't seen it explicitly stated, the
following is implied: down through the centuries, the Aryans and
non-Aryans somehow managed to not interbreed. When Jesus
(an Aryan) came around, most of the Aryans converted to
Christianity, and only the non-Aryans retained the Judaic
religion. However, since they aren't descendants of Abraham, they
aren't real Jews, while the Aryans are. CI probably also holds
that the Pharisees were all non-Aryans, but that's just
extrapolation on my part.
CI has some other strange twists on normal Christianity (as one
might expect). The first is the origin of the non-Aryan races.
Many (probably most) variants of CI claim that the non-Aryan races
are not descendants of Adam and Eve. Rather, God created each
of the races separately, and that Adam and Eve, the ancestors of
the Aryans (also know as "Adamites"), were the last of the
language and tool using bipeds to be created. The problem with
this is the the Great Flood, since (naturally) Noah, his wife,
and his sons and their wives were all Aryan. But it's really no
problem at all, since flood was actually just a localized
phenomenon, not a worldwide one; it was only the Aryan portion of
the world that God wiped out, and all the other races were left
alone. Some sects of CI have taken this multiple creation
theory to mean that Christianity isn't for non-Aryans: the coming
of Christ was necessary because of Original Sin, which people
inherited from Adam and Eve, but the non-Aryans aren't descendants
of Adam and Eve, so they don't need to be saved by Jesus.
The other big twist is the Fall from Grace, the expulsion from
the Garden. Most variants of Christianity hold that the Fall was
caused by Adam and Eve eating the forbidden fruit from the Tree
of the Knowledge of Good and Evil, but not CI! CI holds that
that Genesis was mistranslated, and, looking at the original
Hebraic version of Genesis, uses the roots of the words "Tree",
"Fruit", and "Eat" to figure out what it really means.
There are two different CI versions of Genesis. One is that the
forbidden act was miscegenation, having sex with the other,
non-Aryan races that God created. And since miscegenation is the
one thing that Adam and Eve were forbidden by God, you know it has
to be a pretty bad sin!
The other, more virulently anti-Semitic version of Genesis is
that the Fall was caused by Eve having sex with the Serpent,
a.k.a. Satan. Satan's child by Eve was Cain, and Cain was
father of the false Jews; thus, the people today claiming to be
Jews literally have demonic blood in them. This variant of CI
claims that these false Jews can still be saved by Jesus, but it's
harder for them, since they are even more naturally inclined
towards evil than ordinary humans.