Key concepts in intonation

Pitch

the quality of a sound produced by the rate of vibrations

The physics

If you hit middle C on a piano, the sound you hear will be produced by a string vibrating 262 times a second. That's usually expressed as 262 cps (cycles per second) or Hz (Hertz). Making a piano do that is actually rather a complicated piece of engineering which relies on the fact that longer strings produce lower notes and thicker strings vibrate more slowly. If you cut the string in half, and hit it again, the note will be exactly twice as high.
Your larynx, where you keep your vocal cords, is considerable smaller than the average piano so the effect is achieved by muscles working to tighten or relax the cords across which you are expelling air, lengthening and thickening them. You may not be able to sing terribly well but you do have considerable control over the pitch of sounds that you produce.
The human ear can (in young people) detect sounds ranging from around 20 to 20,000 Hz and you can detect differences of pitch of around 2 or 3 Hz. (Bats and elephants are a different matter.) Given that the average range of human voice is around 160 Hz, that allows us to detect something like 65 variations.

Language pitch
We don't however, have 65 significant variations to contend with. Authorities vary in this respect but the usual consensus is that significant pitch variations are confined to low, middle and high or sometimes as low, middle, high and extra high.

Stress

giving particular emphasis or importance to a sound

Usually, stress is connected to loudness. If you hit a piano key harder, it will produce a louder sound but one which has the same pitch.
In humans, as usual, the situation is a bit more complicated and there are two issues:

If we make an extra effort to expel more air, we will certainly increase the loudness of the sound we produce. However, the extra effort we make usually results in our also tightening (and shortening) the vocal cords and that, of course, will produce a heightened pitch as well. This means that most stressed sounds are also at a higher pitch.
At the same time, speakers usually also lengthen the sound when they wish to stress it.

This results in three variations, not one:

Stressed sounds are louder
Stressed sounds are longer
Stressed sounds are at a higher pitch

Oddly, the evidence appears to be that loudness is not the deciding factor in what the hearer perceives – duration and pitch are more important.
Thankfully, most authorities concur that we only need to distinguish between stressed and unstressed sounds although there is clearly a whole range of intermediate possibilities with the three variables working together.

Rhythm

the flow of words and phrases determined by the relation of long and short or stressed and unstressed syllables

Here is where we bump into a common division into syllable- and stress-timed languages. The argument (based on Abercrombie, 1965) is that some languages (such as French) are syllable timed with each syllable taking the same amount of time to be uttered. So we get:

In French:
french
in which all syllables take the same amount of time to utter and in English:
english
in which each section (called a foot) has one stressed syllable and any number of unstressed syllables, all of which take the same time to utter.

More recent work has somewhat refined the theory, placing languages on a cline from those which are strictly syllable timed to those which are strictly stress timed with gradations in between.

The tone group

Descriptions of intonation are not concerned with the pitch of individual syllables but rather with pitch patterns or tunes. The phonological unit set up to handle the structure of tunes is the tone group
(Brazil, Coulthard & Johns, 1980:6)

The usual division of the tone group or unit is into four bits:
(Prehead) (Head) Nucleus (Tail)

The reason that three of those bits are in brackets is that they may not always occur – the nucleus, however, is always present.
This is where things get complicated because there is less agreement among authorities in the analysis.

Nucleus: This is the key bit to focus on. It is the syllable on which the main pitch change starts. Some people call it the tonic syllable.
Although authorities differ, it's possible to distinguish 4 types of pitch movement on the nucleus: fall, rise, rise-fall and fall-rise. Complicating the issue are the three or four pitch types recognised above (low, middle, high and extra high). Combine all those and it's clear that there are 12 possible variations to consider. The nucleus is followed by the ...
Tail: This, when it is present, continues the pitch movement which occurred in the Nucleus. Some authorities don't even recognise its existence and combine it with the Nucleus or Tonic.
Prehead and Head: Together, these two constitute the syllables up to but not including the Nuclear syllable. Some people don't even recognise two categories here and use Pretonic to describe the whole bit leading up to the Nucleus (or Tonic). Usually, the pitch movement on this part of an utterance can be described in the same way as the pitch movement on the nucleus (fall, rise, rise-fall and fall-rise). The movement, is, however, smoother and less emphatic.
There are those, Brazil et al, for example (1980:9), who suggest that the pitch movement on the Prehead and Head, while variable, is not actually significant. That's convenient for teaching purposes because it means we can virtually ignore it and focus our learners on the pitch movements occurring on the Nucleus or Tonic.

In the guide to intonation on this site, 6 possible pitch changes on the Nucleus are suggested. They are

neutral tone

falling tone

rising tone

sharply rising tone

rising tone followed by falling tone

falling tone followed by rising tone

Intonation and communication

The communicative importance of all this is:

where the speaker decides to place the Nucleus in an utterance and
which of the 6 pitch changes is used

For example, consider the utterance:

she's going to see that film

Say this sentence aloud, doing three things:

moving the Nucleus around from item to item along the utterance
raising the pitch of the Nucleus
changing the pitch movement in one of the four ways: fall, rise, rise-fall, fall-rise

and you'll soon hear the differences.
For example:

Place the Nucleus on the first syllable (she's) and use a sharply rising pitch movement. You'll get a sentence which sounds like an astonished question asking for the hearer to confirm its truth.
Try again with the Nucleus on go: and a rise-fall, you'll get a sentence that sounds as if the speaker is confirming her fixed resolution to see the film.

Try doing those two and then writing down the way you see the intonation using arrows to show pitch movements and writing bits of the utterance above or below a base line to show pitch and then click here for a picture of how it might look.

intonation

Try a few more of these for yourself and identify how the sentence will sound to a listener.

This is the kind of thing which can be presented and practised but do not imagine, however, that our reactions and attitudes are signalled only by the intonation we use. Far from it. If you or your materials imply anything of the sort, your students will be seriously misled. For example, the intonation in the first diagram above could just as easily show extreme anger as astonishment.
The point is:
There are no arguments for teaching intonation in terms of attitude, because the rules for use are too obscure, too amorphous, and too easily refutable. (Brazil, Coulthard & Johns, 1980:120)

Intonation does play a communicative role, of course, but in conjunction with the setting, the roles of speaker and listener, their intentions and their shared information in the setting.

Now you can try a short test or go on to the guide to intonation proper.

Go to the index for the pronunciation section of the in-service guides

References:
Abercrombie, D, 1965, Studies in Phonetics and Linguistics, London: Oxford University Press
Brazil, D, Coulthard, M, Johns, C, 1980, Discourse Intonation and Language Teaching, Harlow: Longman
Nespor, M, Shukla, M and Mehler, J in The Blackwell Companion to Phonology, 2011.