Use of equalizers in communication systems for Intersymbol Interference
A (linear) equalizer tries to compensate for the (linear) filtering effect of the channel. This filtering introduced by the channel creates intersymbol interference (ISI), so the goal of the equalizer is to reduce (or, ideally, eliminate) ISI, such that the symbol error rate is reduced to an acceptable level. Linear filtering creates ISI by smearing out the symbols over more than one symbol period, making neighboring symbols interfere with each other.

Time-domain and frequency-domain equalizers are just specific implementations of equalizers, their basic function is the same. Furthermore, compensating for the impulse response of the channel is the same as compensating for its transfer function because impulse response and transfer function are just two different descriptions of the same phenomenon: linear distortion.

Unknown channels can be estimated, but usually adaptive equalizers are used, which can adapt to unknown and/or time-varying channel characteristics. Usually, a training sequence of known symbols is used for an initial adaptation of the equalizer. There are also methods that do not require a training sequence; these methods are referred to as blind equalization (e.g. constant modulus algorithm).

The output of the equalizer is indeed an estimate of the transmitted signal; the final decision is done by the slicer following the equalizer.