Before learning about estimators, you first need to learn that there is a distinction between "data" and "data generating process." An estimator is an attempt to learn something about the data generating process, by using the data.
Take a basketball player's shots in a game, like this:
Feet Made the basket?
In statistics, you view the Yesses and Nos as coming from processes specific to the feet from goal, like flipping coins that are bent, but with degree of bending related to feet from the goal. Clearly, 0% is not the true measure of the player's ability to make a shot from 10 feet (the "process"), as the data (0/2) seem to suggest. But 0/2 is indeed an estimate.
An "estimator" considers the potentially observable data values, rather than the actually observed data values, like this:
Feet Made the basket?
Say the Y's are coded as 0/1, 0 = Missed, 1 = Made.
The obvious estimator of the player's probability of making the shot from 10 feet is (Y2+Y5)/2. This is a random variable that can take values 0, .5 or 1.0. It is unbiased, but not a particularly good estimate, because we do not think that the player's true ability from 10 feet is either 0, .5, or 1.
Unbiasedness of an estimator is not the most important criterion for judging whether it is "good." Accuracy is more important: You want the estimator to be generally close to the target.
A better estimator would "borrow strength" from the nearby data, by assuming that the probability of making the shot is a continuous function of distance. Logistic regression will do that, giving you a better estimator (despite its bias), in the sense that it tends to be closer to the target than the estimator (Y2+Y5)/2 given above.
Thus, a major benefit of the estimator framework is that it gives you a rational way to decide between estimates, such as the simple average versus the logistic regression estimate referred to above. With estimates, all you have is two numbers. Which is better? There is no way to know. But in the estimator framework, you can compare the distributions of potentially observable values of the estimates, and pick the one that generally tends to be closer to the target.