Literally, without echoes. Anechoic refers to the absence of audio reflections. The closest thing to this situation in nature is the great outdoors, but even here there are reflections from the ground, various objects, etc. It is almost impossible to create a truly anechoic environment, as there is no such thing as a perfect sound absorber. At high frequencies, it is possible to create near-anechoic conditions, but the lower the frequency, the harder this is (Absorption is wavelength dependent. As an example, a 100 Hz wave is about 10 feet long; the absorber must be at least 1/2 a wavelength deep to function properly. It quickly becomes impractical to create a large enough space with enough material in it to absorb low frequencies).
It is not desirable to create anechoic or near-anechoic conditions in a recording studio. The total absence of reflections skews perception, and will not result in good recording or mixing decisions. Anechoic chambers are used for testing and spec’ing microphones and loudspeakers, as well as for a variety of other audio measurements.