You already got two nice answers, but since it still seems unclear for you let me provide one. Likelihood [is defined as][1]

$$ \mathcal{L}(\theta|X) = P(X|\theta)  = \prod_i f_\theta(x_i) $$

so we have likelihood of some parameter value $\theta$ *given* the data $X$. It is equal to product of probability mass (discrete case), or density (continuous case) functions $f$ of $X$ parametrized by $\theta$. Likelihood is a function of parameter given the data. Notice that $\theta$ is a parameter that we are optimizing, *not* a random variable, so it does not have any probabilities assigned to it. This is why Wikipedia states that using conditional probability notation may be ambiguous, since we are not conditioning on any random variable. On another hand, in Bayesian setting $\theta$ *is* a random variable and does have distribution, so we can work with it as with any other random variable and we can use Bayes theorem to calculate the posterior probabilities. Bayesian likelihood is still likelihood since it tells us about likelihood of data given the parameter, the only difference is that the parameter is considered as random variable.


  [1]: http://stats.stackexchange.com/questions/112451/maximum-likelihood-estimation-mle-in-layman-terms