Moods are often intertwined with tense and aspect. In fact, the triad is commonly referred to in linguistics as tense-aspect-mood or just TAM. Aspect and tense are relatively straightforward, but how does mood relates to them exactly?
I'll summarize what Mark Rosenfelder says in Advanced Language Construction, pp. 146-156 (some of which referring back to Mood and Modality, F.R.Palmer 2001).
First we start with modality, which is concerned with the status of the proposition, i.e. "how true it is, and whether it's subject to obligations and intentions". In contrast tense (at its most basic degree, ignoring semantic expansion) is interested to whether it happens before or after now, and aspect with overall temporal relation (hence the possibility of contrast between tenses within a given aspect).
Rosenfelder describes Palmer's classification of modals like so:
- Epistemic: speculative, assumptive & deductive
- Evidential: reported, sensory & deductive
- External: obligative, permissive, commissive
- Internal: abilitive & volitive
(Most discussions bundle all of the "factual" branch as one "evidentials" category because languages with evidentials tend to have only one of the two sub-branches, which is also why deductive is listed twice without being self-contradictory)
Basically working from that, a mood is what happen when modality bundles up with tense, and there it gets a little messy. Terminologically, Rosenfelder believes that indicative/subjunctive and realis/irrealis systems are separated more clearly by geography (specialists in given language families prefer one or the other) than grammatical reality.
In any case, realis/indicative applies protoypically to real events, irrealis/subjunctive to "less real" stuff. What is included in each category may vary considerably (future events and imperative may categorize in either one, for example, depending on individual languages). Conditionals are irrealis in virtually all systems, but in Indo-European linguistics are traditionally treated as a third mood separate from both indicative and subjunctive.
In practice, moods are often more of a semantic division within the verbal paradigm owing to the lack of a separate marker for them (in most European languages I'm aware of, for example). When an explicit irrealis marker exists, it often takes over separate marking for tense and aspect, that is, there will be no overt marking of the subcategories like future vs. imperative (both only irrealis) or past vs. present (both realis). Manam is such a language (how unusual this is is unclear from Rosenfelder's writing).