The following is clearer than anything I have ever heard.
The term anatman
is usually translated as "non-soul", but in reality atman
is here synonymous with a personality, an ego, a self, an individual,
a living being, a conscious agent, etc.
The underlying idea is
that, whatsoever be designated by all these names, it is not a real
and ultimate fact, it is a mere name for a multitude of
interconnected facts, which Buddhist philosophy is attempting to
analyse by reducing them to real elements (dharma). Thus
"soullessness" (nairatmya) is but the negative
expression, indeed a synonym, for the existence of ultimate realities
(dharmata).
Buddhism never denied the
existence of a personality, or a soul, in the empirical sense, it
only maintained that it was no ultimate reality (not a dharma). The
Buddhist term for an individual, a term which is intended to suggest
the difference between the Buddhist view and other theories, is
santana, i.e. a "stream ", viz. of interconnected
facts.
It includes the mental
elements and the physical ones as well, the elements of one's own
body and the external objects, as far as they constitute the
experience of a given personality. The representatives of eighteen
classes (dhatu) of elements combine together to produce this
interconnected stream.
There is a special force,
called prapti, which holds these elements combined. It
operates only within the limits of a single stream and not beyond.
This stream of elements kept together, and not limited to present
life, but having its roots in past existences and its continuation in
future ones - is the Buddhist counterpart of the Soul or the Self of
other systems.
From Theodore Stcherbatsky in 'The Central Conception of Buddhism and the Meaning of the Word "Dharma".'
