Why finiteness
Cognitive linguistics (and many other linguistic, psychological, and philosophical theories) assumes that people perceive the world as objects (i.e. things) and events (i.e. actions or relations). Objects are limited in space while events are limited in time. They are prototypically coded by the basic word classes in language: nouns and verbs, respectively. Typical clausal constructions are formed from a verb that takes nouns as its arguments (1) and (2).
(1) [a noun] likes [a noun]
(2) [The tyrannosaur] likes [latte]
Language is built on the prototypical case that verbs take things, which are coded by nouns, as their arguments. But of course, we all know that tyrannosaurs don’t like drinking latte but bathing in it. [bathes in latte] is not a thing but an event and not a NP but a VP, and (3) is therefore incorrect.
(3) # [The tyrannosaur] likes [bathes in latte]
It must be said that you can also force the main clause to take a clausal complement, i.e. a subordinate clause, (see the first clause in 4). Languages differ in how easily it can be done. Nonfinite verbforms and constructions around them are often described as shortened subordinate clauses. I think that's quite misleading because there is no on-to-one correspondence.
If you however want (or have) to put an event/verb in a noun slot in the clause, you have to use class-changing derivation. Verbs used as verbs in syntax are called finite and verbs used as something else are called nonfinite. (There is a debate if that’s a binary distinction.)
Nouns can of course be coded like verbs too, but in European languages that’s done by adding a light verb, like be, have, use… West-Greenlandic has many productive denominal verb derivations.
The one above is a kind of characterization. Defining finiteness is somewhat problematic. Nedjalkov uses terms decategorization and recategorization, which I find useful. Decategorization means that a verb loses (some of) its "verbiness". Recategorization means that it gains (some) nominal features. Decategorazation and recategorization aren’t always complete. Participles, for example, often have tense, which is a verbal category.
(4) Nedjalkov’s examples of decategorization (D) and recategorization (R). - stands for absence and + for presence.
[-D/-R] Sentential complements (I know that he comes)
[+D/-R] infinitives (I want him to come)
[-D/+R] “clausal nominalizations” (I disapprove his driving the car so carelessly)
[+D/+R] deverbal nouns (arrival etc.)
Edit: [-D/+R] “clausal nominalizations” can be exemplified quite badly in English. Nedjalkov has a better example (4b) from Quechua. The Embedded clause 'Juan was here.' is a complete "finite" clause, but at the end of it there is a nominalizer and an accusative marker that tells that the embedding is the object of the matrix clause. So the clause has not lost any of its "verbiness" but behaves as a noun. If English had it as a clausal nominalization the clause would need about, like think usually needs. "I think about that Juan is here."
(4b)
Ñuka-ka [Juan kay-pi ka]-shka-ta yani.
I-TOP [Juan this-in be]-NZR-ACC think-SG1
I think that Juan was here.
The quartet of nonfinite verb forms (4b)
Ñuka-ka [Juan kay-pi ka]-shka-ta yani.
I-TOP [Juan this-in be]-NZR-ACC think-SG1
I think that Juan was here.
What is here called nonfinite verb forms are those that are decategorized, i.e. are less "verbi".
The typological quartet of nonfinite verb forms is: infinitives, converbs, action nominals, and participles. The two former ones are decategorized but not recategorized, i.e. not “verby” but not “nouny” either. They differ in that the infinitive is a complement of the main verb (or a noun), i.e. a nonfinite complement clause, while the converb is an adjunct, i.e. a nonfinite adverbial clause. That is the clause is incomplete (or elliptic) without the infinitive or a nominal complement, (5) without a complement and (6) with an infinitive. The lack of a converb or a nominal adjunct, on the other hand, does not make the clause incomplete, (7) with a converb and (8) without.
(5) # The tyrannosaur likes.
(6) The tyrannosaur likes to bathe.
(7) The tyrannosaur bathes in latte before smoking a cigar.
(8) The tyrannosaur bathes in latte.
Infinitives and converbs can usually take objects and other arguments like any verb, (7), and (8). Subject coding can differ. It’s often the same as that of the main clause, as the herbivore in (9). The Latin term for that is accusativus cum infinitivo. It can also be coded as an adverb, like for in English. Many languages have several converbs for different adverbial functions: time, manner, instrument, purpose, other action… English uses conjunctions to specify them.
(9) The tyrannosaur forced the herbivore to wait.
Action nominals and participles are decategorized and recategorized, i.e. they get a new word class. Action nominals are deverbal nouns and participles are deverbal adjectives. They behave syntactically like the words of the word class in question. Dependents of nouns are usually genitives, so action nominals take their ‘arguments’ in similar forms, (10). Adjectives describe nouns, and so do participles as well (11).
(10) [the tyrannosaur’s smoking of the cigar]
(11) A smoking tyrannosaur looks silly.
Action nominals can well appear both as complements (11) and adjuncts (12), because so can nouns too.
(12) A puppet saw the tyrannosaur’s bathing in latte.
(13) The herbivore held the towel during the tyrannosaur’s bathing in latte.
Grammars know a bunch of different names for nonfinite verb forms: gerund, gerundive, masdar, supine. They mostly taken from Latin grammars. Their use is very varied in grammars, and if you use one of them in your conlang, you have to explain it much better than a typology-based term. They can be replaced by the quartet.
Nonfinite verbforms in real languages
The four names of the verbforms are typological generalizations, which Haspelmath calls comparative concepts. Real languages don’t read grammars. (The cosmos doesn’t real physics either, but that’s another thing.) So, the real forms in real languages don’t always be as prototypical as those above. English -ing form is especially unpleasant. It can work as infinitive (14), converb (15), action nominal (16), and participle (17).
(14) The girl started smoking (= to smoke) a cigarette.
(15) The girl entered the room smoking a cigarette.
(16) The girl’s smoking irritated me.
(17) A smoking girl entered the room.
A girl smoking a cigarette entered the room.
Turkish has verb forms that function as both action nominals and participles. Finnish has infinitives that inflect in local cases, which is a trace of their past as action nominals. Infinitives often have side functions as purposive adverbials (18), where you would wait for a converb.
(18) The tyrannosaur baths in latte to get a sophisticated colour.
The forms in real languages rarely fit the typological definitions, but a main function can usually be found, and the form named after it. Then you can write clauses like: "Infinitives are used to code purpose." instead of "Supines function as complements of verbs and also code purpose."