The participle is a construction in which a verb is transformed into an adjective. It attributes an action or the concept of an action directly to a noun, instead of having the noun directly carrying out the action. Just like verbs, participles can have tense and voice, although usually the range of tenses available for a participle will be more limited than that for regular verbs. In English we have two participles, the past participle which is either exactly the same as the past tense, or irregular (like speak, spoke, spoken), and the present participle, which is just a regular verb with -ing attached. Latin participles are somewhat more complicated. What causes the complexity is the nature of Latin nouns. Besides having qualities of number and gender, they also decline according to their grammatic function. All adjectives modifying Latin nouns much match in number, gender, and case. Thus, all participles must as well. Despite their adjectival status, participles still retain some of their power as verbs, and can take direct objects while still modifying a noun.

Formation

Present Active
The present active participle is easily constructed. You simply takes the Present Active Singular First-Person form of the verb (always the first to be listed in a dictionary), chop off the -o ending, and insert the root vowel for the conjugation (see Conjugating verbs in Latin). This will give you the present stem. To the present stem you add the ending -ns, unless the present stem ends in -i-, in which case you add -ens instead. This participle applies to all genders. There is no present passive participle to worry about.

Perfect Passive
The perfect passive participle is a little more difficult. There's no systematic process required to derive it, but rather a requirement of memorization. In most cases, especially with first conjugation verbs, the perfect passive participle will be the present stem plus the ending -tus, -ta, -tum for masculine, feminine, and neuter, respectively. This is not always the case, however. For example, the perfect passive participle of video (to see) is visum, not videtus. There are many of these irregularities, especially in common verbs. The perfect passive participle of a verb will always be listed in a dictionary as its fourth principle part. It is best just to memorize this when learning the verb.

Future Active
The future active participle is based off the perfect passive participle; yet another reason to make sure the fourth principle part is memorized. Simply, the -us ending is dropped, and -urus, -ura, -urum is substitued in its place for the respective genders.

Future Passive
Reminiscent of the present active particple, the future passive is formed by taking the present stem and adding endings -ndus, -nda, -ndum for the respective genders. Once again, i-stem verbs will use -endus, -enda, -endum instead.

Declension

Declension of participles according to case and numbers is absolutely essential. Any noun the participle modifies, even if the participle is on the other side of a verb, must be matched up exactly with case and number. That being said, there's a slight contradiction. The present active participle remains the same no matter the number, case, or gender of the noun it modifies. The perfect passive, future active, and future passive participles will all correlate in gender, number, and case. To see how this unfolds, refer to Latin noun declensions.

Uses

Present Active
The present active participle gives a sense of action happening contemporaneous to the main verb. Based on context, it may also be translated with although, when, while, since, or if. When translated with although, the participle will often be accompanied by tamen (nevertheless). Some examples:

Puer ludens maestum sentiebat - The playing boy was feeling sad

Puer ludens tamen maestum sentiebat - Although the boy was playing, nevertheless he was feeling sad

Perfect Passive
The perfect passive participle gives a sense of action happening prior to the main verb. It works much in the same way as the present active participle. One very common use of the perfect passive is in clauses with cum, which give direct sense of with the blah having been blah. Some examples:

Vir territus cursavit - The man, having been frightened, ran hither and thither.

Cum viro territo cursavit - With the man having been frightened, he ran hither and thither.

Future Active
Very similar to the present active, except with a connotation of action subsequent to the main verb.

Rex capturus insulam, victoriam eius amplectit - The king about to capture the island considered his victory

Future Passive
Again, very similar to the perfect passive, except with a connotation of action subsequent to the main verb. There's also a sense of obligation.

Libros excrendos tande legit - He read the books which had to be sorted slowly.


Moreland, Floyd L., Fleischer, Rita M. Latin: An Intensive Course. Berkely: University of California, 1977.
Traupman, John C. The Bantam New College Latin & English Dictionary. New York: Bantam Books, 1995.
Cawley, Kevin. Latin Dictionary and Grammar Aid. http://www.nd.edu/~archives/latgramm.htm