For instance, if you say, "I came here to eat," or "We want something good to eat," you are using the infinitive "to eat" to express reason or purpose. How do translate something like this in Latin?
In Latin, the infinitive is not used to introduce a reason, or "purpose clause" as a Latin grammar would put it. Here are some other options, which I will gear toward the (very broad) use case of English translations using the infinitive with a sense of purpose.
- Ut + Subjunctive
ut means "that, in order that" and introduces a subordinate purpose clause. The negative of a purpose clause is ne. An example:
Loquor ut meminerim.
I speak to remember.
- Ad + Accusative Gerund
Ad pugnandum aggreditur.
He comes forward to fight.
- Genitive Gerund + Causa
Debellandi causa venit.
He comes to conquer.
- Gerundive Participle with object
This one is a bit tricky: if you are speaking of an object on which something "ought to be done," you can modify that word with a gerundive. This applies well to your second example sentence:
Aliquid boni edendum volo.
I want something good to eat.
(This translation might be made better...comments welcome.)
- And my favorite... the supine!
The supine is an oft-disregarded form that, for most intents and purposes, is the same as the neuter singular passive participle. It is quite common in everyday speech (just read Plautus and see it appear every page) and expresses purpose with verbs of motion. This translates your first example well:
I came to eat/praise/applaud.
The answer above is pretty comprehensive! I don't yet have the reputation points to make this into a comment, rather than a full answer, but there are a few things worth adding.
First, In the case of Aliquid boni edendum volo — I think the most literal translation would be: "I desire something of good having-to-be-eaten." boni is then a partitive genitive. I think edendeum is really the future passive participle, since the accusative of the gerund will usually want ad to express purpose, as noted in the answer. But they are morphologically identical.
Also note that the accusative of the supine is most commonly used with verbs of motion (eo, ire, venire, et cetera).
As the point of departure for this question was the English phrase “I came here to eat”, it might be worth mentioning that in this construction “to eat” is not (at least historically) an infinitive, but the preposition “to” with the Old English dative of a verbal noun. Structurally it is thus more like Latin “ad edendum” (preposition plus accusative gerund).
As others have said, the infinitive wasn't generally used to indicate purpose in Classical Latin. But this does happen fairly regularly in later times.
For a famous example, in the apocryphal Acts of Peter, Peter is fleeing persecution in the city when he runs into Jesus. He asks, Domine, quo vadis? ("Lord, where are you going?") and Jesus responds Romam eo iterum crucifigi ("I'm going to Rome to be crucified again").
To add to the other answers. Quo can replace ut to introduce a relative clause of purpose. Compared to the other alternatives, this option seems to be scarcer; it is used primarily (but not exclusively) when the clause contains a comparative:
legem brevem esse oportet quo facilius ab imperitis teneatur. (The law should be short to be more comprehensible to the unskilled)