Going back in history we notice that the hardest aspect of musical notation to get right is the notation of rhythm. Once the staff of lines and spaces was discovered (or invented, traditionally ascribed to Guido of Arezzo), it took another 500 years of experimentation before efficient solutions to the problem of rhythmic notation were discovered.
In music we have different specific words for time. The "timing" of a piece is its duration in minutes and seconds. But we also have the concepts of "beat" or "pulse", of "meter" and of "rhythm" itself. The pulse is the recurring, regular beat that underlies most music. Its speed is indicated in a score with a "metronome mark". The metronome was invented shortly after 1800 and Beethoven was one of the first composers to give specific indications in his scores. The sign for a quarter note followed by an equal sign and a number tells you that there are so many quarter notes per minute. This is usually seen in the top left of the score right after a tempo word. These tempo words are usually Italian (such as Allegro or Adagio), but are often German or French, or even English. They give a rough idea of tempo and mood whereas the metronome mark gives a precise indication.
Another crucial element of the notation of rhythm is meter. We feel the pulses of music in little packages of two, three, four or more beats. Each package is shown in the score with a barline to separate it from surrounding packages. The typical meter of most pop music is 4/4, meaning each measure or bar has four beats. In pop music usually the 2nd and 4th are stressed. This is what is called the "backbeat" because it is actually a variation from the norm. The regular stress in 4/4 is ONE two three four (to show that one is really stressed and three a bit less). Another common meter is 3/4 which is used in minuets and waltzes. Again, the first beat is stressed. Here, let me show you what this looks like. Here are some examples:
This was done in Finale, which can do just about anything. First off we have the tempo word, Adagio, which means "slowly", followed by a quarter note being equal to 60. So the pulse is one beat per second. The first measure is 4/4, with four quarter notes. Note that this is just for example. You can write pretty well any kind of rhythm you like within that basic meter by using a pattern of short and long notes. But it has to add up to four quarters or the equivalent. Next is a measure of 3/4 with just three beats.
Now we get a bit fancy. The next measure is a compound time signature, meaning that the beat-unit is not a simple note like a quarter note, but a dotted note. Each beat contains within it three smaller note values instead of two. This time signature is used a lot in dances such as the gigue. It has rather a lilting feel. 6/8 is called compound duple time because it consists of two beats, each of which is a dotted note. The next measure, 9/8, is compound triple time because it has three beats, each of which is a dotted note.
The next line shows what weird things you can do with composite time signatures. A composite time signature is one that has a variety of beat units, not just quarter notes or dotted quarter notes, but a mixture of, perhaps, quarter and eighth note beat units. The first measure in the second line, with the time signature 1/2 + 5/8 means that each measure consists of a half note followed by five eighth notes or a group of four eights followed by a group of five eighths. The last measure is a quarter beat followed by another quarter beat, then three eights, then a final quarter beat. So those are two examples of composite time signatures. A popular piece using a composite time signature is "Take Five" by Dave Brubeck which is in 5/4 divided into two groups of three quarter notes followed by two quarter notes.
But obviously, no-one would be crazy enough to actually use the two composite time signatures I showed as examples, right? Right? In fact, they are both found in the second movement of my second symphony. I originally had them all notated in 9/8, but I realized that this was concealing the real subdivisions. What you see there, plus a couple of others, is what is actually going on in the meter of the piece!
Let's have a little listen to "Take Five" just to show you how natural a composite time signature can feel.