BASIC MUSIC NOTATION
NOTES
Music is written on sets of five horizontal lines, each one called a "stave" or "staff". Before any music is written onto the stave it has to be set up with certain features of the music to be written. An empty stave might look like:
Apart from the five lines there is another symbol. This is the "clef", which tells us on which line or space the notes are written. The above clef is the "treble clef" or "G clef" and it tells us that the note G is written on the second line from the bottom. Music for smaller instruments and the higher human voices is usually written with this clef.
Notes are named from A to G and as you go from A to G the pitch rises. Once G is reached the notes start again at A. Thus, given that the note G is on the second line up, we can write the following:
The note C at the left hand end is called "middle C" because it is in the middle of the range of notes that people can hear properly. Any two notes with the same name will in one sense sound the same but one will be higher than the other and they are an "octave" apart. An octave has eight notes, from A to A or B to B or C to C etc.. If the notes of an octave are written out in either ascending order as above or in descending order then that is a "scale".
The other common clef is the "bass clef" or "F clef". It is a fancy letter F which shows where the note F is (second line down, between the two dots) and looks like:
The note C at the right hand end is again middle C. A stave with a bass clef is often paired with a stave with a treble clef to give a wide range of available notes. Piano music is usually written like this.
RHYTHM
Notes can be of differing lengths, by which we mean they last for differing lengths of time. The length of the note is shown by whether the round part, or the "head", is filled in or empty and whether the note has a "stem" with some tails. Most music has a beat or pulse to it and the shape of the note says how many beats the note lasts. The following example lists the common ones:
Notice that most notes have a stem. Some stems point upwards and others point downwards. This means nothing and is just a way of saving space between lines of music. Some of the notes lasting 1/2 a beat have a tail and others are joined in pairs (or maybe fours or eights or other numbers) by "beams". Whether music is written with tails or beams is partly a matter of style but can also indicate note grouping during performance.
There are other note lengths. For instance, to get notes
that last a quarter of a beat they would be written with two tails or beams.
Also, it is often required to be able to write notes that are half as long
again as one of the above notes. In this case a dot is placed after the note.
The following are examples: 
Notes of differing lengths have names:
RESTS
Sometimes it is necessary to tell the performer that at a particular time no note should be played. To do this we need "rests". These have the same durations as corresponding notes but no note should be heard:
Here is an example with lots of rests:
CHORDS
Two or more notes can be played at the same time. When written down in music these notes are written in a vertical column:
Each note of the chord is held for the same time. Notice that the sixth chord has one note on the other side of the stem from the others. This is just to prevent overcrowding. Sometimes music is written with chords and other notes on the same stave:
In this case the upper notes are played whilst the lower chords are playing. So, when the first chord is played the upper C and D are played one after another so that these two crotchets take the same time as the minim chord.
BARS
Music is usually divided up into groups of a small number of
beats, usually 2, 3 or 4 but sometimes more. To show these divisions the music
is divided into "bars" or "measures". Vertical lines,
"bar lines", are placed at the end of each bar. For example, the
following is divided into groups of 4 beats: 
Notice that there is a vertical bar line ever 4 beats, remembering that the first bar has a minim in it and the last bar has a semibreve in it. Notice that at the end of the piece there is a special double bar line. Also notice the numbers just after the clef. These make up the "time signature". The time signature has two parts: the upper number tells us how many beats there are in a bar and the lower number tells us whether to regard crotchets, quavers or minims (usually) as 1 beat. Thus if the lower number is a 4 then one crotchet is one beat, if the lower number is an 8 the 1 quaver is one beat and if the lower number is a 2 then one minim is one beat. Other examples are:
A piece of music can start on any of the beats on the bar other than the first. In this case there will be a bar line at the start of the second bar and the last bar will contain sufficient beats to complete the first bar:
Usually, in performance, the first beat of the bar is played a little more strongly than the others and this adds to the sense of pulse.
SPEED
Different pieces of music go at different speeds. The speed is known as the "tempo". For different tempos the speed at which the beats are counted will be different, ranging from about 30 beats per minute for very slow music to 100 beats per minute for steady music and up to 180 (or more) beats per minute for very fast music. In practice it is reasonable to play the music at a convenient speed which captures the spirit of the music.
There are several "tempo markings" which specify approximate speeds. These are written above the music, usually at the start but maybe at other points as well if the speed is meant to change:
Adagio slow
Andante slowish but not too slow
Moderato at a moderate speed
Allegretto not too quick
Allegro quick
Presto very quick
There are many other directions that can be given to indicate the speed and manner of the music.
DYNAMICS
The loudness or quietness of a piece of music is shown by "dynamic markings". These are written above or below the music. Typical dynamic markings are:
pp pianissimo very quiet
p piano quiet
mp mezzo piano moderately quiet
mf mezzo forte moderately loud
f forte loud
ff fortissimo very loud
cresc. crescendo getting louder
dim. Diminuendo getting quieter
In addition to the last two there are "hairpins",
which are pairs of diverging lines (getting louder) or converging lines
(getting quieter). The following example shows some dynamic markings: 
This starts quiet then gets louder in steps in bars 2 and 3. In bar 4 it gradually gets quieter then in bar 5 is gets gradually louder. Finally, in bar 6 it gets quieter to the end.
ARTICULATION
The way that notes are played one after another is called "articulation". Notes can be played in very joined up or very separated ways and individual notes can be emphasized. To show that a group of notes should be played smoothly, without any separation between them, the "slur" is used. This is a curved line draw above or below the notes to be slurred. Two or more notes of the same pitch with a slur between them are meant to be played "tied". This means that the first note is sounded but is held for the combined length of the tied notes.
Notes which are to be played in a detached manner, or "staccato", have a dot drawn above or below them, as convenient. Notes that are meant to be emphasized will have symbols such as ^, - , sf or one of a variety of variations of these drawn above or below them. Another common symbol is the curved line with a dot inside, or "pause". A note with this symbol above or below must be held for longer than written. The following shows some examples:
SECTIONS OF MUSIC
We have already seen that music is divided into bars. Music can also be divided into bigger sections. The end of a section is often marked by a double bar line and sections are often labelled with letters. Whole sections can be repeated and maybe the performer must go back to the beginning when the end is reached. The example below illustrates some of these features:
This is a complete piece of music. It is quick (allegro), with four beats (crotchets) to a bar, and starts moderately loud (mf). Notice the double bar line with two dots after it and the double bar line with two dots before it. The section between these double bar lines should be repeated. Notice also that the last bar of this section has a bracket and the number 1 above it. This means that this bar is to be played the first time only, and when the section is repeated this bar is omitted and the bar marked with a bracket and the number 2 (second line) is played instead.
The second line gets louder half way through (f) and has some big chords. At the end of this line is the direction "D.C. al fine". D.C. means "da cappo", or back to the start, and "al fine" means up to where then end of the whole piece is marked by the word "fine", or finish. This is found in the last bar of the first line and this is where the piece ends, at the double bar line.
This is just a brief introduction to music notation and the interested reader is referred to "The AB Guide To Music Theory" parts 1 and 2, available from all good music shops and some bad ones.