Democracy – or the right to take part in the government, directly or through representatives who have been freely chosen in regular and honest elections that guarantee the equal right to choose - is a human right. In fact, these words have been taken almost literally from article 21 of the Universal Declaration of Human Rights. A more detailed defense of democracy as a human right is here.
The “equal right” part has often been summarized in the phrase “one man, one vote” (nowadays that includes women, fortunately). Democracy tries to give everyone equal political influence and to give equal consideration and protection to everybody’s interests, and it does so first of all by giving everybody equal voting rights. However, “one man, one vote” isn’t enough to give everyone equal political influence and to ensure that everyone’s interests are taken into account. The way that people use their vote can be influenced through propaganda, unfair media attention, unequal education, excessive use of money in campaigning etc. That is why democracies don’t stop at equal and universal voting rights, but also try to improve education and public discourse and impose limits on unequal participation in political campaigning (e.g. campaign financing limits, fair and balanced media coverage etc.).
If some people are excluded from the vote, or if other people have a disproportionate influence on the way people vote, then some of us have more political influence than others, and hence more power to protect our interests. Those who believe in democracy accept that certain people always more power than others to advance their interests, because political talents and interest differ, but the purpose of democracy is to equalize influence as much as possible. That is why nobody has more than one vote (it used to be different in the early stages of democracy) and nobody’s excluded from the suffrage (if you kindly forget about some categories of people; see here and here). And it’s also the reason why asymmetrical influence on the way people use the vote is discouraged. If all votes aren’t equal, and everybody doesn’t have the same rights and means to participate in and influence political decisions, then it’s difficult to claim that the people govern.
One of the mechanisms that change the weight of a vote that I haven’t mentioned yet is one that is build into the election systems of some countries. A notable example is the U.S. Senate.
I will not repeat how the U.S. Senate deviates from the principle of “one man, one vote” (see here if you want, or here). Suffice it to say that each U.S state is represented by two senators, regardless of population. In an interesting but utopian effort, Neil Freeman has kindly altered the map of the U.S. so that all states contain a more or less equal population, and “one man, one vote” is restored:

(source, click to enlarge)
Easier would be to change the composition of the US Senate methinks, or to just abolish it.
And it’s not just the U.S. Senate that fails to respect the “one man one vote” rule. The Presidential election in the U.S. is in fact an indirect election: voters elect members of the electoral college. These members are so-called electors who have pledged to a presidential candidate. These electors then elect the President. Presidential candidates have to have a majority in the electoral college – more than half of the 538 electors. Because states can send a number of electors to the college that doesn’t match the population size of states – some states can send more per capita electors than other states – presidential candidates can get a majority in the college without having a majority among voters. The U.S. Constitution specifies the number of electors to which each state is entitled. Electoral votes are allocated to the states each decade to reflect population shifts, but every state is guaranteed three electoral votes before allocation kicks in.
Proponents of the college argue that the system protects the rights of smaller states. Numerous constitutional amendments have been introduced, unsuccessfully, in Congress seeking to alter the Electoral College or replace it with a direct popular vote.

In Wyoming, there are 143,000 people for each of its three electoral votes. The states with the weakest votes are New York, Florida, and California. These states each have around 500,000 people for each electoral vote. In other words, one Wyoming voter has roughly the same vote power as four New York voters. (source)
More human rights maps here.
51.053562
3.720867