You can now recognise a participle. This chapter is about the harder half: working out what one is doing in the sentence in front of you.
There are three answers, and one question comes first. Ask it every time, before anything else about the form:
Is there an article?
Article present, it is behaving like an adjective or a noun. Article absent, it is usually describing the circumstances of the main verb — usually, because an anarthrous participle — one with no article — can still be attributive. The counting says which pattern is commonest. Participles without an article far outnumber those with one: the nominatives alone come to 3,918, most of them adverbial, against 1,433 articular participles in every case — so the third use is much the largest group, and the one with no signposts.
With the article, beside a noun
The attributive use. The participle modifies the noun, exactly as an adjective would, and it agrees with it in gender, number and case.
εὐρύχωρος ἡ ὁδὸς ἡ ἀπάγουσα εἰς τὴν ἀπώλειαν
"Broad is the road that leads to destruction." Note the shape: article, noun, article, participle — ἡ ὁδὸς ἡ ἀπάγουσα. That repeated article is how Greek usually marks an attributive participle, and it is the construction chapter 6 gave you for adjectives.
Translate it with a relative clause. Not "the leading road" but "the road that leads": English participles will not carry the weight and a relative clause always will.
ὁ λαὸς ὁ καθήμενος ἐν σκοτίᾳ φῶς εἶδεν
"The people who sat in darkness saw a light." Same shape, and again a relative clause is the natural English.
With the article and no noun
The substantival use. Give a participle an article and no noun to modify and it becomes the noun itself, exactly as an adjective does in chapter 6, where οἱ πραεῖς is "the meek".
ὁ πιστεύων εἰς τὸν υἱὸν ἔχει ζωὴν αἰώνιον· ὁ δὲ ἀπειθῶν τῷ υἱῷ οὐκ ὄψεται ζωήν
"The one who believes in the Son has eternal life; the one who disobeys the Son will not see life." Two of them set against each other, with no noun anywhere in either half — the whole verse is carried on them.
The gender tells you what kind of thing is meant, as it did with adjectives: masculine for a person, neuter for a thing or a quality.
μακάριοι οἱ πενθοῦντες
"Blessed are those who mourn." It is one of the commonest constructions in the book.
Without the article
The adverbial use — much the commonest, and the one with no marker at all. The participle is not describing a noun but the circumstances under which the main verb happens.
Translate it with a clause and let the context choose the conjunction: when, after, because, while, by, or simply and.
ἀκούσαντες δὲ ἐβαπτίσθησαν εἰς τὸ ὄνομα τοῦ κυρίου Ἰησοῦ
"And when they heard, they were baptised into the name of the Lord Jesus." Here the hearing plainly came before the baptism — the usual order with an aorist participle, but the aspect does not require it.
Ἰδὼν δὲ τοὺς ὄχλους ἐσπλαγχνίσθη περὶ αὐτῶν
"And when he saw the crowds, he had compassion on them" — or "because he saw the crowds". Greek does not say which, and both are honest. That ambiguity is the normal case, not a problem to be solved: the participle states the circumstance and leaves the relation open. Where you need because rather than when, the sentence has to supply it.
The genitive absolute
One construction worth learning by name. When a participle has a subject of its own — someone other than the subject of the main verb — both go into the genitive, and the phrase stands loose from the sentence. Hence "absolute", from the Latin for "loosed".
Τοῦ δὲ Ἰησοῦ γεννηθέντος ἐν Βηθλέεμ τῆς Ἰουδαίας
"Now when Jesus had been born in Bethlehem of Judea…" — and the main clause that follows is about the magi, not about Jesus. Two different subjects, so the participle goes genitive.
Ἀναχωρησάντων δὲ αὐτῶν ἰδοὺ ἄγγελος κυρίου φαίνεται
"And when they had departed, behold, an angel of the Lord appears." αὐτῶν is the participle's subject; the angel is the verb's.
There are 735 genitive participles in the New Testament, and this is the main reason one turns up in that case without an article. Ask who its subject is: the answer will not be the subject of the sentence.
What to watch for
Find the article first. It is the fastest question you can ask: with one, the participle is adjectival or substantival, and you then ask whether there is a noun for it to modify; without one, it is usually adverbial. Usually, not always — an anarthrous participle can still be attributive, which is why the sentence has the last word.
Check the agreement. It is often the only thing telling you which noun a participle goes with, especially in a long sentence where the two are far apart.
Do not force the adverbial sense. When, because and after are your additions, not the writer's.
And aorist still does not mean past. It means the action is viewed as a whole.
Participles are the last genuinely hard thing in the morphology, and the only way through is meeting a great many — which, at one verb form in four, you will.