'Democracy' is of course a variable word. In the strict sense, democracy is rule by the Demos and based on something like the following principle:
(1) Every citizen, by being a citizen, equally has the right to vote in all matters of legislation.
Naturally, this is not what most people mean by 'democratic' most of the time. In modern times we often talk about democracy in talking about 'democratic elections'. Now, as the ancient Greeks quite correctly recognized, all election systems for legislatures are by nature inherently not democratic but oligarchical, because they are based on the principle of restricting the access of citizens to legislative voting power, and all sufficiently large election systems necessarily create a class of Eligibles, People Capable of Being Elected, in contrast to those who, practically speaking, stand no chance of being elected. The only genuinely democratic method for representative democracy is sortition, choosing by lot, because then it's still genuinely true any citizen can become a representative. There are ways to have oligarchies without elections, but all election systems, introduced into a society, create oligarchies. They are, in fact, quite effective at doing so.
When we talk about 'democratic elections', then, what we really mean are elections with extra concessions that alleviate their oligarchical tendency. A common, and commonly discussed such concession is something like the following principle:
(2) Every citizen, by being a citizen, equally has the right to vote in elections.
This universal suffrage principle is sometimes confused with a very different concession you could have:
(3) Every citizen, by being a citizen, has the right to vote in elections with votes equal in number to every other voter.
You could have the universal suffrage principle without much caring about the equal number principle, and vice versa; the universal suffrage principle is about how votes are made and the equal number principle is about how votes are counted. You could have a voting system in which everyone could vote but some people got one vote, some people got two votes, for any reason whatsoever. Likewise, you could have a voting system in which not everyone equally has the right to vote, but, when they vote, everyone has an equal number of votes. Both of these give you a sense in which you can say your election is democratic, i.e., makes democracy-leaning concessions.
Both (2) and (3) are often confused with yet another principle:
(4) Every citizen, by being a citizen, has the right to vote in elections with a vote of weight equal to every other vote.
This equal weight principle is an independent principle, as it concerns not how votes are made, nor how they are counted, but the effect that they have. You can have (2), (3), and (4) in any combination. But there are problems with (4) that don't arise with (2) or (3), and the primary problem is that whether votes have equal weight is relative to measure, and there are many different measures you can use. Two votes of equal weight by one measure can fail to be of equal weight by another. This just follows from the fact that equal weight is a matter of the consequences.
Weight of vote, to be more exact, is a matter of how your vote contributes to the result in light of every other vote. This depends, of course, on exactly what overall population you are considering; in Presidential elections, even if votes weigh equally in a state, they don't weigh equally across jurisdictions of any states, because they are different elections, albeit for the same office.
In one district I may be voting in a larger population than I would be in another, thus diluting my vote. One could thus say that my vote weighs more in a smaller population than in a larger population. The relative voting population, however, could be actual voters, likely voters (and there are several different ways of measuring who counts as a likely voter), or eligible voters.
If A has a vote in a gerrymandered district clearly designed to favor a given result and B has a vote in a district not so gerrymandered, someone could say that their votes do not contribute equally to the respective results of their districts, and are thus not of equal weight, even though A and B are not in the same voting population and even if the voting populations are the same size.
If people wanting A can easily vote and people wanting B can only vote with difficulty (e.g., due to geographical location in the district), even in the same voting population, do their votes weigh equally? We don't often think about whether vote-weighting should consider difficulty of voting, but nothing about the weight of a vote relate to other votes rules out doing so. It seems that we could see this as a bias in the system that favors a vote for A over a vote for B even if they are counted the same when given, thus increasing the ease of A winning over the scenario in which difficulty of each individual vote is the same across the board.
One thing that often comes up in equal weight discussions is how many candidates there are are, which affects how much the vote captures the preferences of the voter.
One could use other measures, for any number of purposes, but the point is that whether two votes are of equal weight depends on the measure being used, and the measures are not guaranteed to give you the same result. Thus one should not talk about equal weight of votes without specifying a measure of weight and (in practical applications) why that measure is the relevant one to use.