First let's review: meter is how we refer to the organization of the beats in a given piece of music. In order to define the meter for a particular piece, we place a time signature at the start of each line of music. Each type of meter (duple, triple, quadruple) has an implicit stress pattern. For instance, in 2/4 the stress pattern is "strong weak, strong weak," and in 3/4 the pattern is "strong weak weak, strong weak weak." But all meters place the strongest beat on beat one, which is referred to as the downbeat.
Every measure in a piece of music should conform to the number of beats defined by its time signature. This means that every measure in 4/4 should add up to four quarter notes. Any combination of notes and rests may be used, but each measure needs to add up to four quarter notes. However, there is an exception to this rule: the pickup measure or anacrusis.
Some pieces of music do not begin on a strong beat. For example, if you sing "Hap-py Birth-day to you," you may notice that the first stressed beat (the real downbeat) is on the word "birthday," not the word "happy." Since the rule is that the strongest beat in the measure needs to be beat one, then "Birthday" needs to be beat one, and "happy" needs to be a weak beat. The conventional way to do this is to start the song with an incomplete measure called a pickup measure, or anacrusis.
In the example above there are two eighth notes for the two syllables of the word "happy" that lead into the first downbeat on the word "Birthday." In order to count a pickup measure, we simply count backwards from the end of the measure. For example, the pickup in "Happy Birthday" begins on the last two eighths of a 3/4 measure, which would therefore be counted "three and."
If a piece has a pickup measure, it is common practice to make the final measure shorter by the length of the pickup measure, so that the pickup measure and the final measure add up to one complete measure. For example, if a piece in 4/4 has a one beat pickup, then the final measure should contain the remaining three beats (see example 1). This is done in part so that if the piece repeats, the number of beats at the end will smoothly transition back to the beginning.
Here are a few examples of well-known songs that use a pickup measure (anacrusis).