A measure in music notation (also called a bar) contains a certain number of beats as specified by the time signature. They are purely a conceptual way to divide the music into rhythmic, managable chunks. Each measure is indicated by two vertical bar-lines drawn completely through the staff or grand staff from the top line to the bottom one. For the computer-literate, you could loosely think of each note as a bit and each measure as a variable-length byte.
For instance, if the time signature is 4/4, then four quarter notes or their equivalent will reside in each measure. The following example has a staff with three measures in 4/4 time. The first measure has four quarter notes (one beat each), the second has two half notes (two beats each), and the third has one whole note (four beats).
/\
|----| /-------------------------------|------------|------|
| |/ | | |
|----/---------------------------------|------------|------|
| /| 4 * * * |* | |O |O | O |
|--/-|/\---------|-----|-----|-----|---|--|-----|---|------|
| | | | 4 | | | | | | | | |
|-| -|--|--------|-----|-----|-----|---|--|-----|---|------|
| \ | | | | | | | | | | |
|---\|_/-------------------------------|------------|------|
| |
| \|
If a certain note's duration carries it beyond the end of a measure, two different notes at the same pitch are drawn in each measure and a tie is used to connect them.