Beowulf is way of building a supercomputer out of a bunch of smaller computers. The smaller computers are connected together by a LAN (local area network) which is usually an Ethernet. Historically, these smaller computers were cheap computers; even Pentium I and 486 class machines running Linux! That was the appeal. Using cheap hardware that nobody wanted and using it to build something resembling a supercomputer.
It's said that the Beowulf idea can enable a modest university department or small research group to obtain a computer which can operate in the gigaflop range (a billion floating point calculations per second). Normally, only mega rich corporations like IBM, AT&T and the NSA can afford such awesome computational power.
In the taxonomy of parallel computing, a beowulf cluster is somewhere below a massively parallel processor (MPP) and a network of workstations (NOW).