The two arguments to COUNTIF are range and criteria. It will not accept
multiple ranges; it *will* accept multiple criteria.
=COUNTIF(rng1,{criterion1,criterion2}), array entered, or
=SUM(COUNTIF(rng1,{criterion1,criterion2})) entered normally, both work.
Ah, semantics. =COUNTIF(X,{y,z}) applies criteria y and z separately to range X,
producing two separate results. Only in cases where no more than one of those
criteria could be satisfied for each cell in X would this approximate what most
English speaking people would consider multiple criteria - in that limited
context, the sum of the result would be the number of cells in range X
satisfying either y or z. E.g., =SUM(COUNTIF(X,{"<0",">0"})) returns the count
of nonzero numbers in range X. When the conditions aren't mutually exclusive
(for Alan's benefit, that means that the same cell in X could satisfy multiple
criteria), the result isn't useful except as the equivalent of two separate
COUNTIF calls. E.g., =COUNTIF(X,{"<=0";">=0"}) counts zeros in both returned
values, so the sum would double count any zeros in range X.
In general, multiple AND criteria should be expressed as
=SUMPRODUCT(--(X=y),--(X=z))
and *robust* OR criteria as
=SUMPRODUCT(--((X=y)+(X=z)>0))