Participles

Participle is one of the moods of the verbs in Greek. It is an adjectival form of the verb. I.e. although it is based on a verb, it functions similarly to the adjective. Such an adjective characterizes an object as performing the corresponding action. 

Participles have the following grammatical categories: Tense, Voice, Case, Number & Gender. Possible tenses are: Present, Aorist & Perfect. As in the case of the Subjunctive Mood, the tenses don't express a time aspect, but only the kind of the action; i.e. Present corresponds to continuous\repeated action, Aorist to simple actions, and Perfect to completed actions. 

Similar to Adjectives, Participles can be used in three main ways:

• Attributive use. In this case, the Participle is used like an attributive adjective. It has to be in an attributive position, i.e. ὁ λύων ἅνθρωπος οr ὁ ἅνθρωπος ὁ λύων, and it has to have the same Case, Number & Gender as the Noun it modifies. In this case, the Participle can often be translated by the relative clause, specifying the noun, i.e. in the case given the man, who is untying.

• Substantive use. This is similar to the Substantive use of Adjectives, i.e. the Participle function by itself without a Noun. For example, ὁ λύων, would mean the untying one or the one, who is untying. 

• Adverbial (circumstantial) use. Participles, that are used adverbially, are usually Nominative since they are modifying the subject of the sentence. They must be in a predicative position, i.e. appear before the article or after the noun: λύων ... ὁ ἅνθρωπος or ὁ ἅνθρωπος ... λύων. Such participles add a background action that the subject performs in addition to the action given by the main verb. This background action may be performed before or during the main action, it can signify the manner the main action is performed, the reason or purpose behind it, etc. They are usually translated by a subordinate clause. So the sentence λύων λἐγει ὁ ἅνθρωπος, can be translated as the man speaks, while he is untying, or the man speaks, because he is untying, or the man speaks, although he is untying, etc. 

The forms of the participles are given in the following table. For each Tense-Voice combination, one needs to take the corresponding Principal Part, and in the case of Aorist, one also needs to take away the epsilon-augment. In the table, the Nominative Singular and General Singular forms of each of the Participles are given, since they are enough to specify the Declension used. The ending is underlined. You can also find the full tables here.