Steady states are important because we often want to acquire MRI signals quickly. This means we don't necessarily wait for full T1 recovery, or even for full T2 decay to occur before applying the next excitation pulse for the next iteration of a pulse sequence.
Eventually, after a number of exitation pulses (could be 20?, 30?, 100?... it depends) the amount of longitudinal magnetisation just before each next excitation pulse has become constant. The amount of Mz tipped into the xy plane equals the recovery of Mz during TR. In such a case a steady state (SS) has been reached.
How fast a SS is reached will affect an image. For example, what k-space lines are acquired whilst the Mz is still large? What is the SS signal value—low or high?
If we acquire MRI signals so that there is still some residual Mxy when we're ready to start the next TR (short TR compared to T2), we have two choices: get rid of the residual Mxy, or keep it.
Some fast pulse sequences are spoiled which means that they destroy (or cancel out) any residual Mxy magnetisation just before the next excitation pulse, so that Mxy produced by one excitation pulse does does not contaminate the Mxy (and the signal) produced by the next excitation pulse in the next TR.
Another type of fast pulse sequence attempts to undo all dephasing caused by the imaging gradients during the TR, so that a maximum amount of magnetisation is preserved, and keeps transverse magnetisation from one TR to the next. This is called balanced Steady State Free Precession (bSSFP). There are other names for this sequence such as bFFE, TrueFISP and FIESTA.