The first one that comes to mind is the imperfect tense, a type of past tense which doesn't indicate time or completion and implies continuousness. The Romance languages are probably the most familiar languages which use the imperfect, but it's common cross-linguistically. The distinguishing feature of the imperfect tense is that it is "atelic", it does not specify endpoints for an action. Compare this to the simple past tense in English – she ate the food. Ate is "telic", because an endpoint of completion is implied by the word.
English has a way to express the imperfect tense – he was eating the food – but not a separate true conjugation for it. It should be mentioned that all languages have the ability to describe all tense concepts, even if the true tense isn't available in the language. In this case additional words are simply included to provide the intended context. This is exactly what we see in the case of English expressing the imperfect.
There are also rare instances of languages using tense relative to "today," the hodiernal tense. This is subtly different than the typical present tense. Similarly, languages can use a tense referring to events which occurred "yesterday" (hesternal tense) and another which refers to events which will take place "tomorrow" (crastinal tense).
Some languages also use or have historically used a different set of conjugations for literary texts, the so-called "literary tenses". French, for example, uses these. And while these don't represent different concepts of time, these do represent a distinct set of conjugations which represent the context surrounding actions. Namely, that the actions are being narrated or recounted in a journalistic or informational way. In all cases, however, the literary tenses show up in writing rather than in general spoken conversation.
