I kind of understand the meaning of conjugate to something but can't really get the clear picture of the concept. Some video lectures on youtube talk about it but they all seem to just say "Normal Distribution is conjugate to Normal" but for me to understand, it wasn't enough to get it. Hope to hear with least mathematical terms.
Saying it in plain English: if you have some Bayesian model to estimate parameter of interest, then you use some prior distribution for it, likelihood to describe the probability of observing your data conditional on the parameter, so to obtain the posterior distribution. If you have a conjugate prior this means that the prior comes from the same family of distributions and there is a closed-form solution for such problem, so the posterior distribution is directly available. This is exactly the case when you use normal prior for mean parameter of normal distribution. Otherwise you would need some kind of numerical optimization or Markov Chain Monte Carlo to estimate the parameter.