What is petting

Petting means stimulating another person's sex organs or breasts - usually with your hand or mouth.
It's actually an American word - originally meaning to stroke or caress (as one would stroke a 'pet' cat).
There was no equivalent word in 'polite' (British) English, so the US word 'petting' gradually became used worldwide during the 20th Century. In fact, the expression 'petting' and 'to pet' have now become a little old-fashioned - so that in the UK it's now much commoner to speak of:

'touching someone up'
'frigging someone'
'rubbing someone up'
'bringing someone off' (if petting proceeds to climax).

All these are still slang expressions, and the only non-slang English expression is 'masturbating someone'. In practice, this phrase is hardly ever used - because of the fact that most people only employ the word 'masturbation' to mean self-stimulation.
Instead of saying 'petting', many of today's couples talk about 'love play'.
'Heavy petting' is an American expression that usually implies petting below the waist - rather than just involving the breasts.

What does it involve?

The idea of petting is to stimulate your partner's body, in order to give them sexual excitement.
Fortunately, this activity is also usually nice for the person who is doing the petting!
Petting (love play) very often continues until the partner who's on the receiving end actually climaxes.
Bringing someone to a climax by petting is often referred to as:

'bringing them off'
'fetching them off'
'tossing them off' (only used when the recipient is male).

Petting is often referred to in older books as 'foreplay' - meaning that it's something you can do before starting intercourse.
Doing it before intercourse is actually an excellent idea, because it prepares the body for actual penetration.
Indeed, nearly all women need plenty of petting before intercourse - in order to get their genitals 'ready'.
To be specific, petting before intercourse will:

make the vagina open up
make the 'love juices' flow
get the woman excited so that she really enjoys intercourse.

Petting without intercourse?

Sure. Plenty of couples - particularly young ones - go in for petting 'sessions' for half-an-hour or an hour, but don't go on and have intercourse.
These sessions are particularly useful for young people who need to express their emotions and find sexual relief, but who don't want to progress to intercourse (or perhaps can't progress to intercourse because of their circumstances).

What happens when a woman 'pets' a man?

She can do any of the following:

stroke the man's penis
rub his penis
kiss his penis
lick his penis
suck his penis - this is often described as a 'blow job' (but please do not blow - this is dangerous!)
use a vibrator on him (though in practice vibrators are much more commonly used by men on women).

What happens when a man 'pets' a woman?

He can do any of the following:

caress her breasts and nipples
kiss them
lick them
suck them
stimulate her clitoris, vulva and vagina with his fingers
stimulate these areas with his lips or tongue
use a vibrator (or other sex aid).

Risk of catching STDs or getting HIV?

There is a very small risk of acquiring infection through petting, but it is much less than with full sex. Statistically, the biggest risk at the moment is of catching herpes, through being given oral sex by someone who has a 'cold sore' on their lips. There is also some risk of catching gonorrhoea ('clap') through oral love play.
The danger of catching HIV is believed to be very low, but is not zero. There has been at least one case in the USA in which oral petting led to the transfer of the HIV virus. For a full discussion of the risks of oral sex please see the complete article on this subject.

Conclusion

Generally, petting is great fun and very safe. Learning to do it skillfully is well worthwhile - especially as skilled love play is essential for helping most women to reach a climax.