What are two limitations of democracy?

Democracy has various critics from numerous different and sometimes diametrically opposed political orientations. Traditionally, the political spectrum is divided into left and right, but another common dimension articulated by political scientists is the vertical dimension, or authoritarian and libertarian. So it is possible to have libertarian right-wing politics, authoritarian (or anti-democratic) left-wing politics, and vice versa.
It is important to note that there are multiple forms of democracy—for instance, social democracy found in Scandinavian countries, a more direct form of democracy in Switzerland, and a democratic republic found in the United States. There are also pseudo-democracies in certain authoritarian states which only maintain the illusion of being democracies. There is also the economic factor to consider in assessing a democracy—whether it is freely capitalist as in the United States, or more regulated as in certain countries in Europe, or contains certain elements of socialism or social democracy.
So, when speaking of the limitations of democracy, there are always a number of factors to be considered. Strictly speaking, democracy is simply "the rule of the people." This is much more complicated in a modern nation with millions of citizens and sophisticated forms of democratic representation.
Using that definition of democracy as the primary form to assess its limitations, however, there are basic conclusions which one might draw. Because democracy almost always depends on some form of majority rule, this always has implications for political minorities and individual interests. Depending on a number of social factors, this majority force can be manipulated by certain political actors or trends—for instance, demagoguery or what is called "mob rule," a form of group think.
While democracies can be very flexible, they can also become unwieldy in their convoluted forms of representation. However, this is perhaps mostly a problem of large representational democracies, where the processes of election, legislation (particularly legislation, which can become very tiresome), and accountability become involved in bureaucratic labyrinths; at the same time, of course, this can be a counter to more direct forms of democracy (which rely more heavily on referendums, plebiscites, and the like), which may be too impulsive and reactionary.
There are many issues involved in democracy; however, the alternatives based in minority or elite rule depend on an inherent lack of freedom and upholding of privilege.

The two most prevalent limitations to democracy are the rule of the masses, and its course to totalitarianism.
To start, the rule of the mass has been one of the most fundamental concerns when entertaining the idea of democracy. Democracy is, in theory, a one-man one-vote mode of governance. That means everyone within a body politic has the exact same amount of agency as everyone else when deciding policy. Within a body politic there is a disparity of resources, and inevitably that disparity is the many at the bottom have little, and the few a the top have a lot more. If we assume that the primary incentive of voters in a democratic society is economic, that is, the want to obtain and carry out policies which will produce economic benefit for the voter, and if we assume that as a democracy which has as a condition that the majority will get there way, than it follows that the majority of people who have little are going to vote together on policies that will benefit their current economic standings within society. This is not in and of itself a problem, but the problem arises when the policies that the masses (the majority) prioritize are more often than not policies that are antagonistic to efficiency in many sectors, but the mass itself becomes a tyrannical majority. (That is, in theory, democracy is the only mode of governance where the majority can vote the minority into slavery). Couple that with the incredible irrationality of the voter, the psychological fact that when people get into groups they are much more easily persuaded, and an overall lack of education and critical thinking in how politics works, and you have a scary herd of people controlling the whole of society.
Secondly, and this arguably is the most insidious of the two, is ideological tyranny. Democracy must have a free press (and free speech in general) to function. It is the one mode of governance that needs free flowing ideas as a precondition. The issue is that it is almost part of the human condition that we want to suppress alternative ideas, especially if those ideas are perceived as harmful. And based upon the first limitation, the mass rule, once a mass gets into power their first inclination is to suppress alternative, antagonistic, and contradictory ideas. Because they are the mass within democracy they have the ability to do so. They impose a de facto tyranny of ideas. Without the ability for alternative ideas to generate democracy crumbles into an ochlocracy (mob rule) first, and then becomes purely totalitarian where only one idea is allowed, all others must be snuffed out. (Consider the French Revolution and the Reign of Terror as an example of the consequences of a true democracy; it took a tyrant to save France and democracy from cannibalizing itself.)
Democracy has many limitations, and all are reasons why very rarely have we seen historically and presently a pure democracy. Most prefer a Republic in order to maintain order and structure, while some prefer a much more authoritative approach to governing.

There are multiple kinds of democracies such as representative democracies (such as the U.S. - where officials are elected or somehow chosen to represent the people in a society) and direct democracies (where every member of the society has a direct say in governance- early Athens is an example of a direct democracy).
If we are discussing a representative democracy, such as the United States, one limitation is how faithfully representatives are able to work in the interest of their constituencies. Senators may vote in congress for a bill that the community they represent is not in favor of. This could be due to a number of factors, including that a vocal minority communicates a position to the senator that is actively harmful or not in the interest of a less-vocal majority. Another case might be an issue where a constituency is divided: perhaps if half of a constituency is wealthy and the other half is low-income, the positions of each group would be different though they would be represented by one individual. This is to say that the limitation is that one individual may not be able to represent the needs of a large group of people with different interests.
Another limitation is voting practices and access. In the U.S., many states have one day to vote for elected officials and require certain identification. If community members are unable to leave work, get childcare or do not have transportation or I.D., they cannot show up to the polls and therefore the person elected is not representative of the full community they are claiming to represent. This is a limitation when a representative democracy claims to represent the needs of all people in the society.

One limitation of democracy is that the right to vote requires no special skills or abilities. With few exceptions, all adults are able to vote, irrespective of their knowledge of the relevant issues. It was traditionally thought that the right to vote required wisdom and expertise, and should therefore be severely restricted. If the ignorant and ill-informed were allowed to vote, it was feared, anarchy would prevail. While such concerns seem somewhat overstated in the modern era, they nonetheless contain a kernel of truth. Most people in democratic societies are, by any standards, less than knowledgeable when it comes to often complex political issues. This can often lead to people not really knowing what they're voting for, with often disastrous consequences.
Another limitation of democracy is the status of those who hold minority opinions. In a democracy, decisions are based on the will of the majority. But where does that leave the minority? Decisions based on the will of the majority may very well be democratic, but that doesn't automatically mean that they're right. For instance, in the United States the vast majority of white Southern voters supported blatantly racist and discriminatory policies for many years. In one sense, this was democracy in action. Unfortunately, it also led to the perpetuation of a fundamentally unjust system which kept millions in a state of oppression.