What is the difference between a consistent estimator and an unbiased estimator? What is the difference between a consistent estimator and an unbiased estimator?
The precise technical definitions of these terms are fairly complicated, and it's difficult to get an intuitive feel for what they mean. I can imagine a good estimator, and a bad estimator, but I'm having trouble seeing how any estimator could satisfy one condition and not the other.
 A: Consistency of an estimator means that as the sample size gets large the estimate gets closer and closer to the true value of the parameter.  Unbiasedness is a finite sample property that is not affected by increasing sample size.  An estimate is unbiased if its expected value equals the true parameter value.  This will be true for all sample sizes and is exact whereas consistency is asymptotic and only is approximately equal and not exact.
To say that an estimator is unbiased means that if you took many samples of size $n$ and computed the estimate each time the average of all these estimates would be close to the true parameter value and will get closer as the number of times you do this increases.  The sample mean is both consistent and unbiased.  The sample estimate of standard deviation is biased but consistent.
Update following the discussion in the comments with @cardinal and @Macro: As described below there are apparently pathological cases where the variance does not have to go to 0 for the estimator to be strongly consistent and the bias doesn't even have to go to 0 either.
A: To define the two terms without using too much technical language:

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*An estimator is consistent if, as the sample size increases, the estimates (produced by the estimator) "converge" to the true value of the parameter being estimated. To be slightly more precise - consistency means that, as the sample size increases, the sampling distribution of the estimator becomes increasingly concentrated at the true parameter value.


*An estimator is unbiased if, on average, it hits the true parameter value. That is, the mean of the sampling distribution of the estimator is equal to the true parameter value.


*The two are not equivalent: Unbiasedness is a statement about the expected value of the sampling distribution of the estimator. Consistency is a statement about "where the sampling distribution of the estimator is going" as the sample size increases.
It certainly is possible for one condition to be satisfied but not the other - I will give two examples. For both examples consider a sample $X_1, ..., X_n$ from a $N(\mu, \sigma^2)$ population.

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*Unbiased but not consistent: Suppose you're estimating $\mu$. Then $X_1$ is an unbiased estimator of $\mu$ since $E(X_1) = \mu$. But, $X_1$ is not consistent since its distribution does not become more concentrated around $\mu$ as the sample size increases - it's always $N(\mu, \sigma^2)$!


*Consistent but not unbiased: Suppose you're estimating $\sigma^2$. The maximum likelihood estimator is $$ \hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^{n} (X_i - \overline{X})^2 $$ where $\overline{X}$ is the sample mean. It is a fact that $$ E(\hat{\sigma}^2) = \frac{n-1}{n} \sigma^2 $$ which can be derived using the information here. Therefore $\hat{\sigma}^2$ is biased for any finite sample size. We can also easily derive that $${\rm var}(\hat{\sigma}^2) = \frac{ 2\sigma^4(n-1)}{n^2}$$ From these facts we can informally see that the distribution of $\hat{\sigma}^2$ is becoming more and more concentrated at $\sigma^2$ as the sample size increases since the mean is converging to $\sigma^2$ and the variance is converging to $0$. (Note: This does constitute a proof of consistency, using the same argument as the one used in the answer here)
A: If we take a sample of size $n$ and calculate the difference between the estimator and the true parameter, this gives a random variable for each $n$. If we take the sequence of these random variables as $n$ increases,  consistency means the both the mean and the variance go to zero as $n$ goes to infinity. Unbiased means that this random variable for a particular $n$ has mean zero.
So one difference is that bias is a property for a particular $n$, while consistency refer to the behavior as $n$ goes to infinity. Since Another difference is that bias has to do just with the mean (an unbiased estimator can be wildly wrong, as long as the errors cancel out on average), while consistency also says something about the variance.
An estimator can be unbiased for all $n$ but inconsistent if the variance doesn't go to zero, and it can be consistent but biased for all $n$ if the bias for each $n$ is nonzero, but going to zero. For instance, if the bias is $\frac 1 n$, the bias is going to zero, but it isn't ever equal to zero; a sequence can have a limit that it doesn't ever actually equal.
