A vector is a mathematic representation of some geometric form. If you remember back in math class, (either Algebra or Geometry), your teacher most likely gave you a function and then had you draw it on an x/y axis. This would be a basic example of a vector-based graphic. Programs such as Adobe Illustrator, InDesign, and Corel Draw are all natively vector-based applications that allow designers to create artwork where the resulting file is a collection of vectors. Some examples of vector-based file extensions are .AI, .CDR, .EPS, .PDF, etc.
Vector files are resolution-independent—thus, their resolution is determined only by the output device. Because vector elements are mathematically-defined, scaling (enlarging or reducing their size) simply requires modification of their component mathematical descriptions.