The morphology of verbs can be as rich as that of nouns;
- Tense — The relative time that the action takes place.
- Aspect — The duration and frequency of the action.
- Mood — How the action being discussed relates to reality.
- Agreement — Whether the verb "agrees" with other parts of the sentence, and if so, with what parts?
Every speaker of English should be familiar with verb tenses. They show when (relative to the current time) the action is taking place. Tense is traditionally split into three distinctions: past, present, and future.
However, real languages don't always divide time up in the same way. For example, a language can have a distinction between the far-past and the near-past, or it could do away with distinguishing the present and the future on the verb — It may surprise you to learn that the latter is exactly what English does; English verbs inflect for past and nonpast, and the so-called "future tense" is formed with an auxiliary verb ("will" or "shall") plus the nonpast verb form. Some languages even do away with tense inflection altogether, showing time when relevant with (e.g.) adverbs or adpositional phrases.
Aspect is the inflection of a verb to show how the action is distributed in time. It includes, most basically, a distinction between perfective (completed) and imperfect (incompleted). Other aspects, such as the habitual (occurring regularly over a period of time) and the iterative (done repeatedly), may be found in some languages. Many languages conflate tense and aspect in verbal morphology, so a verb has forms for imperfect past tense vs. perfect past tense.
To give you a better feel for this, in English, verbs are normally imperfect but can be made perfect by turning it into a passive participle and adding the auxiliary verb "have" to the sentence: Contrast "Tom ate his lunch" and "Tom has eaten his lunch". The habitual aspect is more subtle, but can be most easily seen in the distinction between the so-called simple present and the complex present tenses: Compare "The penguins eat fish" and "The penguins are eating fish". Other aspects have no really good construction in English and so we often have to rely on implication and adverbial phrases to get the point across.
Mood (sometimes also called mode) shows a verb's relationship to reality and includes things like the indicative (it happens as stated) vs. the subjunctive (it might happen, it is desired to happen) or imperative (it is commanded to happen). Some languages treat negation (it doesn't happen) or questions (is it happening?) as mood inflections on verbs.
Some languages use moods where we use modal verbs (e.g. can, may, will etc.), for example where we say "I would drink", the Spanish say "(yo) bebería", from "beber". Where we say "He wants to drink", Sanskrit says "pípāsati". And where we say "it is likely that he speaks", Finnish expresses it with "puhunee".
Some languages indicate how the speaker knows the verb is happening as a kind of mood, called evidentiality ("I know because I saw it", "I know because I'm guessing it will happen", or "It's hearsay" etc.). Conlangs with extensive evidentiality marking include Láadan and Qþyn|gài.
Agreement is when a verb is given markings that refer to some detail from another part of the sentence. Most forms of agreement focus around the subject of the sentence but the verbs of some languages will mark details of the object (usually in addition to the subject).
Many languages inflect the verb for the person. Latin and Spanish, for example, have different verb forms depending on whether the subject is first person, second person, or third person, which is reflected on the last syllable or two of the word (hablo "I speak", hablamos "we speak", etc.) whereas English does this only for the third person singular ("he runs" versus "they run"). In languages where verbs inflect for person, it is often possible to omit pronouns altogether and still be understood; in some such languages, subject pronouns are normally omitted, and only included for special emphasis.
Some languages may include gender-marking on verbs instead of/as well as person marking. For example, in Russian, the "past tense" is formed by using a form ending in -l, -la, -lo, depending on the gender of the subject (while in the "present tense" the verb agrees with the person of the subject and not with the gender). In Arabic, most tense forms indicate the person, number (singular, dual, or plural) and gender of the subject.
Some languages (e.g. Basque, Georgian, Quenya) mark the verb for the person not only of the subject but of the object. Thus "I have found it" in Quenya becomes u-túvie-nye-s "HAVE-FIND-I-IT". One language even has forms for more than one subject "I and he", "he and they", for example.
Combining the types of inflection
A language may inflect the verb for all of these things or none of them. Typically, isolating/analytic languages, of course, don't inflect their verbs much. Synthetic languages, the fusional and agglutinative languages, tend to inflect their verbs to varying degrees with these features.
Some inflect heavily in one area but not another. Japanese, for example, inflects verbs not just for tense, but for politeness, desire, mood, and various other things. The verb miru (to see) becomes mimasu to show politeness and minai to show negation. These forms can be inflected further, so mimashita is the past tense form of mimasu and minakatta is the past tense form of minai. However, Japanese verbs show no inflection whatsoever for person - there is not even an equivalent for the minimal -s inflection of English verbs in the 3rd person singular of the present tense.
Polysynthetic languages usually inflect the verb to an incredible degree. In such languages — like some of those spoken or formerly spoken by people native to the Americas and arguably, modern spoken French — the verb may be marked for the number, noun class, and person of both the subject and object, allowing the listener to determine which noun in a sentence is playing which role from the inflections on the verbs alone. This is another way, besides noun cases, for free word order to be possible.
Verbs can be categorized by how many arguments they take, that is, how many nouns are involved by the verb. For example in I am writing a letter to you, the verb write has a valency of three because there are three things involved: the letter; you; and me, the subject. There are also verbs which have a valency of zero, as in It rains. In English, there still has to be a subject, it, but the sentence doesn't define what it is. so it is called "the grammatical It". In some languages, a verb with a valency of zero would simply appear by itself with no noun at all.
This is also not a new inflection of verbs, but a way of categorizing verbs by what arguments they take. Or in other words, how many direct/indirect objects other than the subject are involved in the action. It tells the difference between the letter and you in I am writing a letter to you. and also describes why it sounds wrong to say I am writing you to a letter.