Inter Symbol Interference (ISI) / Equalization in Digital Signal Processing
If the channel just adds noise and does not cause any distortion (as in the first figure), you can design the transmit filter $g(t)$ such that, when combined with its matched filter at the receiver, the Nyquist criterion for zero intersymbol interference (ISI) is satisfied. If the channel adds linear amplitude and phase distortion (as in the second figure), then generally the Nyquist criterion will not be satisfied and you will observe ISI, which will increase the error probability.

A linear equalizer is designed to (at least partly) compensate the effect of the channel $h(t)$. There are different optimality criteria, such as the zero-forcing criterion you mentioned, which eliminates ISI completely, but which does not take into account the noise. A zero-forcing equalizer basically inverts the channel response, if this is possible at all (with a causal and stable filter). If the noise is not negligible, it is usually better to use a different criterion such as the minimum mean square error (MMSE) criterion, which jointly minimizes the effects of ISI and noise.

Since many channel responses are unknown and/or time-varying, i.e., the impulse response $h(t)$ changes over time, equalizers must be adaptive. In practice, most adaptive equalizers use a sort of least mean squares (LMS) adaptation algorithm because it is relatively simple and robust.

This topic is quite broad and complex, so I would recommend that you read a good introductory text on digital communications, instead of gathering unrelated pieces of information from the internet.

Also take a look at this related question and its answer.