[summary: Given a piece of evidence $~$e$~$ and two hypothsese $~$H_i$~$ and $~$H_j,$~$ the likelihood ratio between them is the ratio of the Likelihood each hypothesis assigns to $~$e$~$ For example, let $~$e$~$ = "Mr. Boddy was knifed", and say that Professor Plum is 25% likely to use a knife while Mrs. White is only 5% likely to use a knife. Then the likelihood ratio of $~$e$~$ between the hypotheses "Plum did it" and "Mrs. White did it" is 25/5 = 5/1. See also Relative likelihood.]
Given a piece of evidence $~$e_0$~$ and two hypothsese $~$H_i$~$ and $~$H_j,$~$ the likelihood ratio between them is the ratio of the Likelihood each hypothesis assigns to $~$e_0.$~$
For example, imagine the evidence is $~$e$~$ = "Mr. Boddy was knifed", and the hypotheses are $~$H_P$~$ = "Professor Plum did it" and $~$H_W$~$ = "Mrs. White did it." Let's say that, if Professor Plum were the killer, we're 25% sure he would have used a knife. Let's also say that, if Mrs. White were the killer, there's only a 5% chance she would have used a knife. Then the likelihood ratio of $~$e_0$~$ between $~$H_P$~$ and $~$H_W$~$ is 25/5 = 5, which says that $~$H_P$~$ assigns five times as much likelihood to $~$e$~$ as does $~$H_W,$~$ which means that the evidence supports the "Plum did it" hypothesis five times as much as it supports the "Mrs. White did it" hypothesis.
A likelihood ratio of 5 denotes relative likelihoods of $~$(5 : 1).$~$ Relative likelihoods can be multiplied by odds in order to update those odds, as per Bayes' rule.