The Adapt button automatically adjusts the parameters of the MCMC algorithm in an effort to increase the rate of convergence. Each time you press the Adapt button, information contained in the MCMC sample already collected is used to adjust the parameters of the MCMC algorithm. Then the MCMC sample already collected is discarded, and sampling starts all over again.
The Adapt button is disabled when the parameters of the MCMC algorithm do not need adjusting, and also when too few MCMC samples have been collected to allow choosing new parameters.
For the random walk Metropolis algorithm, the Adapt button adjusts the tuning parameter in order to achieve a satisfactory acceptance rate. At the time of this writing, the target range for the acceptance rate is between 0.2 and 0.4.
For Hamiltonian MCMC, the Adapt button adjusts the leapfrog step size and the number of leapfrog steps in order to achieve a satisfactory acceptance rate. At the time of this writing, the target range for the acceptance rate is between 0.98 and 0.998.