These functions round x to the nearest integer, but
round halfway cases away from zero (regardless of the current rounding
direction, see
fenv(3)),
instead of to the nearest even integer like
rint(3).

For example,
round(0.5) is 1.0, and
round(-0.5) is -1.0.

POSIX.1-2001 contains text about overflow (which might set
errno to
ERANGE, or raise an
FE_OVERFLOW exception).
In practice, the result cannot overflow on any current machine,
so this error-handling stuff is just nonsense.
(More precisely, overflow can happen only when the maximum value
of the exponent is smaller than the number of mantissa bits.
For the IEEE-754 standard 32-bit and 64-bit floating-point numbers
the maximum value of the exponent is 128 (respectively, 1024),
and the number of mantissa bits is 24 (respectively, 53).)

If you want to store the rounded value in an integer type,
you probably want to use one of the functions described in
lround(3)
instead.

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