Time Signatures
Meter is indicated in music notation with time signatures, which consist of two numbers: a top number, and a bottom number. These numbers establish the relationship between the beat and the written rhythms.
The bottom number of a time signature stands for a rhythmic value - 
means half note,
means quarter note, 
means eighth note, and so on (see
Rhythm for more details). The top number indicates how many of this
designated note will constitute a measure. That is, a 
measure contains
two quarter notes, a 
measure
contains two half notes, and so on.
So, what is the relationship between these groupings and the beat? To answer this, first we must discuss simple meter and compound meter.
Simple Meter
In simple meter, the duration of note indicated by the lower number will almost always corresponds with the
beat. For example, in 
time
there are three quarter notes in a measure, and the conductor (or metronome) will beat each of those three
quarters. To put it another way, if the orchestra were to play only quarter notes, they would play one
quarter note for each click on the metronome or beat of the conductor's baton.
As a second example, in time,
the conductor's beat would correspond with the half note instead, and there would be only two clicks per
measure. In this case, if the orchestra were to play only quarter notes, they would play two for each
beat. Thus, if the conductor were to beat exactly the same tempo (beats per minute) as he had for 
time, the piece would
essentially go twice as fast.
Simple Meter
Compound Meter
In compound meter, things get a bit more complicated. In a measure of 
(a common
compound meter), there are six eighth notes. However, a conductor will almost always beat only two
beats per measure instead of six. Thus, time is analogous to 
time. Perhaps became the
standard instead of because there is no convenient numerical representation for a
dotted quarter note. Whatever the case may be, this same convention applies to all compound meters. 
is the same as 
,
is the same as 
, and
so on.
You may have noticed that if 
(dotted quarter) is the unit of the beat,
then each beat will contain three 
(eighth notes), making them
feel like triplets. This is the fundamental difference between simple and compound meter: in simple meter
the division of the beat is duple (two equal parts), whereas in compound meter the division of
the beat is triple (three equal parts).