Participles

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There are two sorts of participles in English.

Present participles

Present participles are the words here in red:
    I am spending more time on it
    Having read the book, I returned it to the library
    After opening the door, he crept silently in
    Entering the museum, he was filled with awe.

Present participles in English are always regular. They all take an -ing ending (but not all words ending in -ing are participles).
The only issue is that we drop the -e at the ending of verbs such as choose when we make it choosing.

Past participles

Past participles are the words here in black:
    I have spent the cash
    The window was broken
    I got the car repaired.
    She hadn't been to New York before

Past participles of regular verbs such as repair are formed by adding -d or -ed. Irregular verbs such as break make the participle in a variety of ways, as broken, in this case. For more, see the guide to lexical verbs.

Present participle functions

Present participles have the following main functions:

To show that an action is progressive	He is writing a new document, She's out swimming at the moment
To show that two actions occurred simultaneously	I heard John laughing, He caught me stealing apples, I spend too much time travelling
To show that one action followed another very quickly	Putting up his umbrella, he left the restaurant
As adjectives	The lecture was unbelievably boring, It was a fascinating story
To explain a reason	Having no money for the fare, he walked to the party

Some notes:

If you have done the exercise on Tense and Aspect then the first row is familiar to you. Other examples include progressive perfect forms such as
They will have been driving for hours
The second and third rows are very similar in meaning and it's often only our knowledge of the world that tells us whether an action was simultaneous or subsequent. In
Opening the door, he crept in
the creeping clearly has to follow the opening but in
Arriving at the party, he saw me
the sense could be that he arrived first and saw me shortly afterwards in the party itself or that he arrived and saw me waiting outside at the same time.
When participles are used as adjectives they cause some confusion for learners. More later on this.
The final row contains examples many feel are rather formal or literary and it's true that they are infrequently used in spoken language. We might prefer something like
Because he had no money ...

Past participle functions

Past participles have the following three main functions:

To show that an action is in the perfect aspect	He has written a new document, She's has swum the Channel
To show that an action is in the passive voice (in all tenses)	The window has been broken, The car is being towed away, The match was abandoned
As adjectives	The students were unbelievably bored, The fascinated listeners were on the edge of their seats, Depressed, he left early.

Participles as adjectives

Compare these:

The book is interesting	I'm interested in ancient history
Everyone is depressed	It's a depressing story
It was an exciting film	The children got over-excited

Can you make a rule for the meaning of -ed adjectives and -ing adjectives? Click here when you have one.

The general rule is that -ing adjectives refer to what a thing or a person is and -ed adjectives refer to how something (or someone) feels.
E.g., We can refer to what something was:
the flight was frightening
or to how someone felt
I was frightened

One thing to note here is that it is often not possible or difficult to distinguish between a passive form and a past participial adjective. When we put the adjective before the noun, it's easier:
The passenger was frightened by the flight (passive use)
and
The frightened passengers hated the flight (adjectival use)
Last note: it doesn't matter how many syllables these adjectives have, they never take the -er or -est endings so we have most bored, not *boredest etc.

Participle vs. gerund

A gerund, simply put, is a verb acting as a noun. For example
I enjoy reading
a gerund: the verb read is acting as a noun vs.
I enjoy books (plural noun)
A problem in English is that the language uses the -ing ending for both participles and gerunds. It is sometimes important to know which is which but the story is quite complicated. There is a separate guide to gerunds and infinitives here.

Related guides
gerunds	for a guide focused on this form and its relation to the infinitive
the infinitive	an essential guide to this related form
tense and aspect	to help you understand how participle verb forms work
adjectives	for a much more technical (and longer) guide to adjectives in general, including a section on participles as adjectives