Since the ma-infinitive and da-infinitive have been frequently discussed here, I'm creating a thread for this specific topic to make those posts easier to find. I'll also add some more information.
Basic info: Estonian has two types of infinitives, known as the ma-infinitive and the da-infinitive.
The ma-infinitive is typically the form that is used as the headword in dictionaries (i.e., the "dictionary form" of the verb).
The ma-infinitive always ends with -ma.
andma = to give
käima = to walk
laulma = to sing
lugema = to read
minema = to go
panema = to put
rääkima = to speak
tegema = to do
tulema = to come
vastama = to answer
Technically, the ma-infinitive is an illative form of the verb. It is therefore used with certain verbs of motion, and also used with verbs that indicate that something is going to start. It is used after the following verbs of motion: istuma [to sit], jooksma [to run], minema [to go], panema [to put], tulema [to come; when used as a verb of motion], sõitma [to travel by vehicle/animal]. The ma-infinitive is also used with the following verbs: hakkama [to begin], juhtuma [to happen], jääma [to remain], kutsuma [to call/invite], kõlbama [to suit/befit], pidama [to have to, must], sundima [to force], viima [to take/lead to]õpetama [to teacher[, õppima [to learn]. The ma-infinitive is also used with certain expressions such as: valmis olema [to be ready], harjunud olema [to be accustomed to], nõus olema [to agree], sunnitud olema [to be forced to], osav olema [to be knowledgeable/good at].
Läheb magama = He/she goes to sleep.
Istume sööma = We sit down to eat.
Tulen aitama = I come to help.
Olen valmis aitama = I'm ready to help.
Pean töötama = I have to work.
Hakkame minema = Let's get going (literally: let's start to go)
The da-infinitive is the other form and it is used in most other circumstances that would require an infinitive in English. Many dictionaries list it after the ma-infinitive as part of the entry.
It does not always end with -da. It can end with -da, -ta, a doubled consonant (based on the consonant in the verb stem) followed by -a, and a few other endings, depending on the word:
anda = to give
käia = to walk
laulda = to sing
lugeda = to read
minna = to go
panna = to put
rääkida = to speak
teha = to do
tulla = to come
vastata = to answer
The da-infinitive is basically used in the situations not covered above for the ma-infinitive, which would require an infinitive in English.
Some examples are: with verbs expressing emotion: armastama [to love], kartma [to fear], meeldima [to like], vihkama [to hate]. It is also used when expressing possibilities or desires: katsuma [to try], lootma [to hope], otsustama [to decide], soovima [to wish], tahtma [to want]. And it is used with certain impersonal expressions, such as on vaja [it's necessary], on tarvis [it's necessary], on aeg [it's time], and so on. For the uses of da-infinitive, this is not a comprehensive list (there can't really be a comprehensive list); as I mentioned above, it's basically used in all situations requiring an infinitive, that are not covered by ma-infinitives above. It is generally easiest to learn the uses of the ma-infinitive by memorization and then use the da-infinitive in the situations that don't require ma.
Examples for da-infinitive:
On aeg minna = It's time to go.
Vihkan seda öelda = I hate to say this.
Mulle meeldib lugeda = I like to read.
Mul on vaja töötada = I have to work.
Tahame laulda = We want to sing.
The verb tulema is a special case because it can require the use of the ma-infinitive or the da-infinitive depending on how it is used. If it is used as a verb of motion, it requires the ma-infinitive:
Tuleme sind aitama. = We come to help you.
But if it is used as an impersonal expression meaning "[someone] has to", it requires the da-infinitive:
Meil tuleb sind aidata. = We have to help you.
This is the way it is generally taught to English-speakers and others learning Estonian as a foreign language. Native speakers don't learn it by the same rules, but learn what "sounds right" and, on the assumption that native speakers already know which form "sounds right," materials for native speakers tend to focus on how other grammatical forms are formed from these infinitives (from the ma-infinitive there is also a -mas and -mast form; from da-infinitive there is also a -des form). Those forms might be a topic for another thread! But their existence means that native speakers learn to focus on the illative meaning of the ma-infinitive, which is generally ignored entirely in texts for speakers of other languages. Ainurakne, as a native speaker, has explained those aspects in much more detail in some of the posts below so rather than trying to summarize that, I'll just post links to those posts.
Previous threads on related topics:
ma- and da- infinitives
ma- and da-infinitives, continued
-nud/-tud, -ma/-da (eesti keeles)
when to use the ma-infinitive
verbs that require the ma-infinitive
(some) verbs that require the da-infinitive
da-infinitive without et
relative past-present-future formed using ma-infinitive forms (inessive, illative, elative)
more about inessive, illative, elative verb forms with -ma