What are The Spies?Are they just children that keep an eye out on their parents to see if they have done anything wrong?

No, the Spies are not just the kids who inform on their
parents.  All kids are supposed to do that.  Rather, the Spies are a youth organization
run by the Party.  It appears that a person becomes a Spy at a young age and then later
graduates into the Youth League.  So the Spies are supposed to be something like the
Hitler Youth -- a group that is meant mainly to indoctrinate the young in the ways of
the Party.

You can see hints about this in Chapter 2.  We
are told that the Spies have been hanging banners for Hate Week.  We also see the
Parsons' son, who is nine years old, wearing his Spies uniform.  This shows us that the
Spies are a formal group that do things other than spying on their
parents.

Here's a line from Chapter 6 that tells shows the
function of the Spies:

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By careful early conditioning, by games and cold
water, by the rubbish that was dinned into them at school and in the Spies and the Youth
League, by lectures, parades, songs, slogans, and martial music, the natural feeling had
been driven out of them.