The yeas and the nays is simply a term referring to how the members of a governing body, such as a congress or parliament, voted. It indicates who votes for or against a specific proposal. Those voting may say either "yea" or "nay" on the issue, and some may even vote "present."
Specifically, in the United States Senate, the term can mean a number of different things. A senator, for example, my prefer a roll call vote on a measure being considered. In most cases, this is so that a record of a particular senator's vote is written down. In calling for a roll call vote, he or she is asking for a clear record of who voted yea and who voted nay.
Although a request may be granted at any point in the Senate in the interest of moving things along, 20% of the senators present must ask for the yeas and the nays before a roll call vote is required. Again, this rule is only in the U.S. Senate. In some legislative bodies, the vote may be called by a much smaller group of individuals. In some places, even one individual is able to demand a roll call vote.
It should also be noted that asking for the yeas and the nays is not the same as calling the question. Calling the question means limiting, and in many cases has the effect of ending, the debate on a certain issue. Instead, all asking for the yeas and the nays does is assure that when debate does end, all individual votes will be recorded.
In some cases, the call happens on pieces of legislation that are virtually assured of passage under any circumstances. Some votes a senator makes may not be very popular in his or her home state, however. Senators may want to leave the option open that some other candidate in their party can use the vote as a campaign issue. Therefore, the calling of the yeas and the nays can be a product of political motivations, rather than of a concern to get the vote counted correctly.