The wikipedia page claims that likelihood and probability are distinct concepts.


  In non-technical parlance, "likelihood" is usually a synonym for "probability," but in statistical usage there is a clear distinction in perspective: the number that is the probability of some observed outcomes given a set of parameter values is regarded as the likelihood of the set of parameter values given the observed outcomes. 


Can someone give a more down-to-earth description of what this means?  In addition, some examples of how "probability" and "likelihood" disagree would be nice.

The answer depends on whether you are dealing with discrete or continuous random variables. So, I will split my answer accordingly. I will assume that you want some technical details and not necessarily an explanation in plain English.

Discrete Random Variables

Suppose that you have a stochastic process that takes discrete values (e.g., outcomes of tossing a coin 10 times, number of customers who arrive at a store in 10 minutes etc). In such cases, we can calculate the probability of observing a particular set of outcomes by making suitable assumptions about the underlying stochastic process (e.g., probability of coin landing heads is $p$ and that coin tosses are independent).

Denote the observed outcomes by $O$ and the set of parameters that describe the stochastic process as $\theta$. Thus, when we speak of probability we want to calculate $P(O|\theta)$. In other words, given specific values for $\theta$, $P(O|\theta)$ is the probability that we would observe the outcomes represented by $O$.

However, when we model a real life stochastic process, we often do not know $\theta$. We simply observe $O$ and the goal then is to arrive at an estimate for $\theta$ that would be a plausible choice given the observed outcomes $O$. We know that given a value of $\theta$ the probability of observing $O$ is $P(O|\theta)$. Thus, a 'natural' estimation process is to choose that value of $\theta$ that would maximize the probability that we would actually observe $O$. In other words, we find the parameter values $\theta$ that maximize the following function:

$L(\theta|O) = P(O|\theta)$

$L(\theta|O)$ is called the likelihood function. Notice that by definition the likelihood function is conditioned on the observed $O$ and that it is a function of the unknown parameters $\theta$.

Continuous Random Variables

In the continuous case the situation is similar with one important difference. We can no longer talk about the probability that we observed $O$ given $\theta$ because in the continuous case $P(O|\theta) = 0$. Without getting into technicalities, the basic idea is as follows:

Denote the probability density function (pdf) associated with the outcomes $O$ as: $f(O|\theta)$. Thus, in the continuous case we estimate $\theta$ given observed outcomes $O$ by maximizing the following function:

$L(\theta|O) = f(O|\theta)$

In this situation, we cannot technically assert that we are finding the parameter value that maximizes the probability that we observe $O$ as we maximize the PDF associated with the observed outcomes $O$. 
