A moving average is a smoothed-out trend line built from an underlying set of data - stock prices in the context of investing. The moving average is calculated by taking, for each point in time, the average values of the underlying data from the previous N timepoints. So, for example, the 10-day moving average for a stock price is the average price of the previous 10 days' closing prices for every given day. The average is said to be "moving" because the time period over which the average is calculated is always the previous ten days of data for each day, and therefore is different for each day.
Moving averages smooth out data over time, and technical traders believe they reveal an underlying trend in a security's price. By increasing "N" - IE, by looking at, for example, a 200 day moving average instead of a 10 day one, a trader would see a trend on a larger scale than with the 10-day moving average, but with decreased responsiveness - IE, only long after large changes in the stock price would the 200-day moving average start to tend in a new direction, given the weight of all the historical prices in the 200-day moving average.
Exponential moving averages give more weight to the latest data points in the moving average calculation, making them more responsive to recent changes in stock price.
Technical traders will look to the convergence or divergence of moving averages, known as MACD, with different periods (or values of "N") to indicate patterns of resistance or support which technical traders believe indicate where a stock price is likely to move in the future.