Legislatures are bodies of government composed of a number of members whose main job is to elaborate and pass legislation. When considering legislature in a national context, we speak of the Parliament, Congress, or National Assembly, and the legislation passed by these authorities are laws. While the organization and form of election of members of legislature may vary depending on the organization of each country, in democratic systems the members are elected, either directly or indirectly, by citizens of each country.
World-wide, legislatures are organized in chambers or houses, and are generally either unicameral or bicameral. When legislature is unicameral, members are organized in only one chamber, in which laws are proposed, discussed and passed or vetoed. In bicameral systems, members are organized in two chambers, each of which generally have certain veto powers regarding law proposals coming from the other chamber, and they also have privileges in proposing certain types of law projects.
The number of members of each legislative body will also vary depending on each country, but it is usually proportional to the corresponding population. In general, this means that legislature will function with an important number of members from different political parties or movements, and each of them have the power to vote on all the subjects that are discussed in forum.
While each member is independent in the deliberation that takes place, they also generally belong to a political party and have been elected on the basis of the political agenda they represent (and, as spiregrain points out, on their personal qualities as well). As a consequence, members of legislature of a same political party generally vote in the same sense in specific topics. For example, members of a conservative political party will all usually vote against law proposals like ones allowing same-sex unions.
While all of this might not be new to those of you who enjoy watching C-SPAN and the likes (I'm guilty of doing the same with the Paraguayan equivalent), the most interesting part is what actually happens behind the scenes. Ensuring that 50 or 60 members of legislature vote in the same sense, even when belonging to the same political party, requires quite the orchestration.
This is where the role of the "whip" comes in. In politics, a "whip" is known as a member of a determined political party (who is usually a member of legislature as well, though not always) in charge of ensuring coherence and discipline within members of the same political party. This means ensuring members attend to sessions where issues of importance to the party are being discussed, and that they vote according to party mandates.
The term "whip" comes from the "whippers-in" in hunting, who are in charge of keeping packs together and preventing them from straying. In this sense, the party whip does the same thing to prevent members of legislature voting against party mandates, and ensuring that enough votes are had to pass or reject proposals that might be in favor or against interests of the party.
Depending on the country, disobeying instructions given by party whips could have different consequences. Depending on the issue, members of legislature that "stray" might be even be expelled from their parties. However, in other parts of the world the role of the "whip" doesn't officially exist, and this discipline is enforced by party leaders, or even leaders of different factions within legislature that might belong to the same party. The discipline, however, is always (somehow) enforced.
*whipcrack!*