A simple moving average takes the mean of the last n closes and recalculates each period. An exponential one weights recent prices more heavily, so it turns faster and lags less, at the cost of more false turns.
The 50-day and 200-day averages are watched far more than any statistical property justifies, simply because so many people watch them. The crossing of one by the other has its own widely reported names, the golden cross and the death cross, and the reporting itself moves flows.
Every moving average lags by construction, because it is made of prices that already happened. It describes the trend well and turns late at every top and bottom, which is exactly the wrong moment to be late.