Relative likelihoods express how relatively more likely an observation is, comparing one hypothesis to another. For example, suppose we're investigating the murder of Mr. Boddy. The suspects are Miss Scarlett and Colonel Mustard, and it just so happens that we think Miss Scarlett is twice as likely to use poison as Colonel Mustard. If we observe that Mr. Boddy was poisoned, this would give us relative likelihoods of (2 : 1) for the hypotheses that Miss Scarlett did it vs. that Colonel Mustard did it.[1]
Relative likelihoods may be given between many different hypotheses at once. Given the evidence = "Mr. Boddy was poisoned", it might be the case that Miss Scarlett, Colonel Mustard, and Mrs. White have the respective probabilities 20%, 10%, and 1% of using poison any time they commit a murder. In this case, we have three hypotheses — = "Scarlett did it", = "Mustard did it", and = "White did it". The relative likelihoods between them may be written (20 : 10 : 1).
In general, given a list of hypotheses the relative likelihoods can be written where it is understood that the list doesn't change if you multiply every term in the list by a positive constant. For example, the relative likelihood list (20 : 10 : 1) denotes the same relative likelihood list as (4 : 2 : 0.2), which denotes the same likelihood list as (60 : 30 : 3). This is why we call them "relative likelihoods" — what matters is their values relative to each other.
Any two terms in a list of relative likelihoods can be used to generate a likelihood ratio between two hypotheses. For example, above, the likelihood ratio to is 2/1, and the likelihood ratio of to is 20/1. This means that the evidence supports the "Scarlett" hypothesis 2x more than it supports the "Mustard" hypothesis, and 20x more than it supports the "White" hypothesis.
Relative likelihoods summarize the strength of the evidence represented by the observation that Mr. Boddy was poisoned — under Bayes' rule, the evidence points to Miss Scarlett to the same degree whether the absolute probabilities are 20% vs. 10%, or 4% vs. 2%.
By Bayes' rule, the way to update your beliefs in the face of evidence is to take your prior odds and simply multiply them by the corresponding relative likelihood list, to obtain your posterior odds. See also Bayes' rule: Odds form.
This could be true if their respective probabilities of using poison were 20% versus 10%, or if the probabilities were 4% versus 2%. What matters is the relative likelihoods, not the absolute magnitudes.