Logistic regression models estimate the log odds of the outcome occurring vs the log odds of the outcome not occurring for a given independent variable (predictor variable). These log odds ratios are not probabilities: instead, they are functions of the probabilities (p) and should be accurately stated in the summary of results. Odds ratios are commonly reported in summaries, and are the exponentiated versions of the log odds ratios.
Ordinal logistic regression requires more careful interpretation. Log odds ratios are calculated for the predictor just as in logistic regression, but are also calculated for (the intercept of) each level of the outcome variable (which typically has 3+ ordered categories in ordinal regression). These intercept log odds ratios reflect the change in the log odds associated with membership in a different level of the outcome variable compared to either the highest or lowest category. There is a strong assumption associated with this model: that the relationship between the predictors and each of the levels of the outcome are proportional (and therefore have the same odds ratio interpretation for the predictor). When there is an ordered outcome variable, ordinal logistic regression is often used, especially if the assumption above holds, this model is informative.