What exactly does it mean to equalize?

What does the term equalize mean? And when can such a term be used?

In My 60 Memorable Games, game 2

It means that the position is equal or roughly equal.  Neither side has an advantage, or any tangible advantage.  

Most generally, I *believe* it means that Black has achieved an equal game without having to submit to defense from White's first move initiative, especially with regard to White's space advantage (as suggested by Harding's book), but more specifically it almost always means that Black has been able to advance a center pawn the full two squares to challenge the center without unnecessary risk. Which center pawn equalizes depends on which opening it is.

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(p. 7)
   From the above it is clear that
there are two types of defence
which Black may adopt in his
search for equality. The first is
the strong point method, where he
retains a Pawn at e5 come what
may. The second is the counter-
attack, where he relinquishes his
e-pawn but compels White to give
up his e-pawn as well, or to weaken
his position otherwise. (It is worth
noting that the execution of this
play does not involve hitting at
the e-pawn on every move; it is the
set-up as a whole that counts.)
   This analysis holds good if
White continues "theoretically" or
"according to Hoyle". Against less
regular lines Black can and should
do what his opponent has
neglected: advance ... d5 and
secure the favourable pawn skel-
eton (1) for himself. In fact, it may
be adopted as a good working rule
that once Black succeeds in playing
... d5 without any immediate harm-
ful consequences he has equalized.

Fine, Reuben. 1989. The Ideas Behind the Chess Openings, Algebraic Edition. New York: Random House, Inc.

(p. 3)
   Black has two methods--one
simple, one complicated--of nullify-
ing the dangers of a space disadvan-
tage. The simple method is to bring
about a freeing advance of a centre
pawn: in the openings with 1 e4 e5,
the freeing move is ... d5; in 1 d4 d5
openings, it is ...e5. In openings
with 1 c4, it can be either move!

Harding, Tim, and Leonard Barden. 1976. Chess Openings for the Average Player. Mineola, New York.