What's the use of the ripple in the Equiripple method and their effect in the filtering process?
There is ripple to three different degrees depicted in this plot:



ripple is the variance that the filter gain is from the target gain.  in the above, there are 3 different frequency segments with 3 different target gains (well, two target gains are the same), and three different error weighting factors.

first frequency segement, from 0 to 0.2 Nyquist: target gain $-\infty$ dB gain (linear gain = 0, a "stopband"), about -67 dB maximum stopband gain (ripple = 0.00045).

second frequency segment,  from 0.2 to 0.6 Nyquist: target gain 0 dB (linear gain = 1, a "passband"), about $\pm \tfrac38$ dB deviation from 0 dB (ripple = 0.44).

third frequency segment, from 0.6 to 1.0 Nyquist: target gain $-\infty$ dB gain (another stopband), about -47 dB maximum stopband gain (ripple = 0.0045).

you will see that the ripple (in linear error terms) times the weighting (100, 1, 10) is about constant.  

ripple is error.  it is the deviation of gain in the actual FIR filter from what your set target is.  this error with each band, besides being a tradeoff between each other (given a fixed number of FIR taps), it is a tradeoff with the steepness of the transition bands.  at least this is the case with the choice of FIR design method.  if the total "average" error is what is salient, then your design method might be least-squares (firls() in MATLAB) and if the maximum error (in each band) is what is important, then the design method is likely Parks-McClellan (firpm()).

now, theoretically, windowed FIR design using a good window (like Kaiser) can sorta compete with the two "optimized" and iterative methods above and there will also be a measure for ripple, but usually the ripple viewed as linear gain, is the same in the passband as in the stopband, but in dB is more in the stopband.  you can hardly see any passband ripple using a Kaiser-windowed design.  and the Kaiser-windowed designed FIR is maybe just a little longer than the optimal Parks-McClellan design, given the same transition band and stopband attenuation.

The "optimal" P-McC design will show ripple (whatever you allow in the tradeoff) in the passband and that might make the Kaiser-windowed design look a lot better to you.  Sometimes i pick a simple Kaiser-window design over P-McC, particularly if i want to upsample by just a factor of two.  In that case, if it's a windowed-sinc, i need not compute half of my output samples because i am copying them from the input (the P-McC does not have that feature).  but if i am upsampling by 4 or more, the P-McC will have shorter FIRs, enough so to make up for the requirement to compute the output sample instead of copying.