In Finnish, verbs have basically four modes: indicative (indikatiivi), imperative (imperatiivi), conditional (konditionaali), and potential (potentiaali). The first two correspond to English modes rather well, whereas the conditional somewhat resembles the subjunctive; and the potential, with no direct counterpart in English, is used to express uncertainty in literary language. This document discusses some details of these modes as well as some additional forms that are sometimes presented as modes.
Morphologically, Finnish uses suffixes for forming the modes (Finnish "modus", from Latin, or "tapaluokka"):
There is an online service, Verbix, for conjugating verbs in several languages; you can use it to conjugate Finnish verbs.
The indicative is used, as in so many languages, to express not only a statement (claim) about facts but as indicating feelings, opinions, and beliefs. Moreover, it can be used effectively as an imperative, e.g. Nyt kyllä lopetat! 'Now you will really quit!'.
The indicative is also used in hypotheses and conditions, as in English, e.g. jos kirjoitan tämän kirjeen... 'if I'll write this letter...'.
Basically, the indicative is a "generic mode", used for different purposes.
The imperative is used, as in English, to express a wide range of wishes, instructions, and commands. It is also used in prayers, where it would be a blasphemy to regard it as a command. A real semantic imperative, a direct command, is in fact fairly rare. The morphological imperative is perhaps most commonly used in instructions ("press the button..."), where it is by no means an order, just information about what one should do to achieve a particular effect. In mathematics, for example, the third person imperative is used hypothetically; a statement like "olkoon n luonnollinen luku" 'let n be a natural number' is not a command in any sense but a hypothesis, a postulate.
Quite often commands are expressed using the indicative and some auxiliary verb, e.g. sinun pitää kirjoittaa 'you must write'. On the other hand, the imperative is used in neutral instructions especially in technical contexts, like kirjoita tähän nimesi 'please write your name here'.
It is worth noting that Finnish seldom uses anything that corresponds to the English "please" even in polite requests. A phrase used in some types of polite requests is ole hyvä, literally 'be good'. More commonly, a question in the conditional is used when politely asking for something. For example, when asking for a bill in a restaurant, you would normally say something like "Could I get the bill, please?" in English, whereas in Finnish we might say Saisinko laskun?, literally "Would I get the bill?", and normally with no question-like intonation.
Sometimes the word optatiivi, optative, is used about some verb forms, like antaos 'oh, please give', even regarding them as a separate mode. I'd say it's more appropriate to consider them as variants of the imperative, e.g. antaos as a poetic form of anna. However, there is no compelling reason to give the optative the status of a mode. In any case, the optative is very rare and only used in poetry.
The conditional is very often used in contexts where English uses the auxiliary verb "would", including cases where no condition is involved, just a polite request. But in particular, the difference between "if I'll write about this..." and "if I wrote about this..." is expressed in Finnish using the indicative and the conditional.
The conditional also expresses a tentative question: "olisiko niin, että..." corresponds to "could it be so that...". Similarly, polite requests most often use the conditional: "voisinko saada..." corresponds to "could I get...". This might be seen as real semantic conditional with an implied condition "if it suits you" or "if it's not too much trouble".
The potential is often described as being a mode of probability. Such usage is however literary only, and basically an attempt to use a simple counterpart to the Swedish expression torde followed by an infinitive. In normal speech, you'll seldom hear any potential. In old popular language, however, the potential might appear in suggestive questions, like "osanneeko hän...", which doesn't actually mean "does he probably know how to..." but rather "I wonder if the actually knows how to...". I have discussed this topic in Finnish in my essay-like document Lieneekö potentiaalikaan tarpeellinen?
In the potential, the verb olla 'to be' has the special stem lie-, so that the forms are lienen, lienet, etc. The third person singular, lienee, often appears in the contracted form lie in everyday language and in poetry. And that form is often used as an adverb, meaning just 'probably, maybe', though this is regarded as incorrect.
A combination of the potential and the conditional, with the morpheme -neisi-, is sometimes called "eventiivi", with the obvious semantics of combining conditionality and probability. And, in fact, such forms occur in Kalevala, the Finnish national epos, in verses like "Tuosta sulho suuttuneisi, mies nuori nuristuneisi." (Kalevala, poem 23). It has been reported that similar forms appear in dialects of Estonian. Panu Mäkinen has noted, in a Usenet message, that Aarni Penttilä has presented the form in his book Suomen kielioppi.
In any case, such forms are not used in contemporary Finnish at all. Theoretically interesting as they might be, there's little need for them.