DBF stands for "dBASE file". It's database stored in a file (usually with .dbf extension). Sometimes it uses two or more files (*.dbt to store big values, *.ndx for indexes, and many others). Every *.dbf file has single table of data. The table has one or more columns (up to 128 or 256, or even more, depending on dBASE version) and data records. dBASE understands simple datatypes (such as text, binary data, number, float, boolean and others, depending on dBASE version).
dBASE was initially started by Ashton-Tate company. Later it was bought out by Borland and finally it was sold to dBASE Inc., which is maintaining it till present day (current version is dBASE Plus). Through all this time some compatible (in most parts) clones of dBASE were created, such as FoxBASE, Clipper and FoxPro.
Major versions are: