The participle is the workhorse of Greek prose, and it is where most readers of Greek finally get stuck. It is worth knowing what you are taking on.
There are 6,640 participles in the New Testament — 24% of its verb forms, one in four. That is more than the infinitives, subjunctives, imperatives and optatives put together. Greek writers build sentences out of participles the way English builds them out of subordinate clauses, and there is no reading the New Testament around them.
So they get two chapters. This one is about recognising a participle when you see one. The next is about working out what it is doing.
A verbal adjective
A participle is an adjective made out of a verb, and it keeps the properties of both.
From the verb it takes tense and voice, and it can take an object.
From the adjective it takes gender, number and case, so it agrees with whatever it belongs to.
Five pieces of information in one word, which is why a participle can do the work of a whole English clause. ἀκούσας means "having heard" — aorist, active, nominative, singular, masculine — and whoever it agrees with is the one who heard.
English has participles too, but only two, and fills the gaps with helping verbs: loving, being loved, having loved. Greek has a full set and uses them constantly.
Tense is aspect, not time
This is the most important thing in the chapter, and it follows from chapter 16: the augment marks time, and a participle has no augment.
So a participle's tense gives you aspect — how the writer is viewing the action — and time only in relation to the main verb.
Present participle: imperfective. Usually the action is going on at the same time as the main verb.
Aorist participle: perfective. Usually the action happened before the main verb.
Perfect participle: a standing result.
"Usually" is doing real work there: the relation is a tendency, not a rule. What is not a tendency is that an aorist participle does not mean "past" — it means the action is viewed as a whole.
The future participle exists and can be ignored: twelve in the New Testament.
The present participle
Two shapes to know, and you have met the endings already.
Active: λύων, λύουσα, λῦον — genitive λύοντος. It is 3-1-3, the pattern from chapter 18: third declension masculine and neuter on the stem λυοντ-, first declension feminine.
Middle and passive: λυόμενος, λυομένη, λυόμενον — an ordinary 2-1-2 adjective, and the easiest forms in the chapter.
The two commonest are λέγων (178) and λέγοντες (150), which between them introduce most of the direct speech in the Gospels.
ἦλθεν ὁ Ἰησοῦς εἰς τὴν Γαλιλαίαν κηρύσσων τὸ εὐαγγέλιον τοῦ θεοῦ
"Jesus came into Galilee preaching the gospel of God." κηρύσσων is present, so the preaching runs alongside the coming — and it takes an object, τὸ εὐαγγέλιον, exactly as a verb would.
The aorist participle
First aorist active: λύσας, λύσασα, λῦσαν — genitive λύσαντος. 3-1-3 again, on λυσαντ-, and the σα is visible.
Second aorist active: the second aorist stem with present participle endings — ἰδών from ὁράω, ἐλθών from ἔρχομαι, λαβών from λαμβάνω. Exactly the arrangement chapter 7 described for the indicative.
Aorist passive: λυθείς, λυθεῖσα, λυθέν — genitive λυθέντος. The θη of chapter 15 is there, shortened to θε.
ἰδόντες δὲ τὸν ἀστέρα ἐχάρησαν χαρὰν μεγάλην σφόδρα
"And when they saw the star they rejoiced exceedingly with great joy." ἰδόντες is aorist, so the seeing came first, which is why English wants "when they saw" rather than "seeing".
The commonest aorist participle in the book is ἀποκριθείς, 94 occurrences: aorist passive in form, from a verb traditionally called deponent, and it means "answering".
The perfect participle, and the shapes to know on sight
The perfect is less common than the other two and easy to spot, because it carries the reduplication from chapter 10.
Active: λελυκώς, λελυκυῖα, λελυκός — genitive λελυκότος.
Middle and passive: λελυμένος, λελυμένη, λελυμένον.
The middle and passive is much the commoner: 448 occurrences against 225, and γεγραμμένον, "written", is the one you meet most.
One form deserves naming on its own: ὤν, οὖσα, ὄν, the participle of εἰμί, "being", genitive ὄντος. It is 3-1-3 like the rest.
So, the endings worth carrying in your head. -μενος is very nearly a guarantee: 444 of the 446 words ending that way are participles, the exceptions being λιμένος and ἀρχιποίμενος. -οντ-, -σαντ- and -θεντ- are strong hints and not guarantees — -οντ- is also in ἄρχοντος, ὀδόντος and every numeral from τριάκοντα up.
Recognising the form is half the work. The other half is the next chapter.