Philosophical republicanism is a relatively newish philosophy, with some roots in ancient Greek, ancient Roman, and Renaissance thought. The central tenet of republicanism is that political structures should be arranged to protect people from domination. Domination is defined as being subject to arbitrary interference.
Pettit motivates the case for republicanism in part by claiming to have found a counterexample to the liberal conception of freedom. Liberals, he says, claim that a person is free just in case they are not interfered with (in various ways). Different liberals put a slightly different spin on this definition of freedom, but Locke, Kant, Mill, Berlin, and others all seem to agree.
But, Pettit says, consider the hypothetical situation in which a slave is owned by a kindly and neglectful master. The master has the right to interfere with the slave at will, for instance, by ordering the slave to work, physically mutilating the slave, or whatnot. But the master simply ignores the slave and allows the slave to do whatever they want. As a result, suppose the slave leads a life much like yours. (If you want, imagine that the law allows slaves to own property, hold jobs, etc., so long as the master allows that or does not expressly forbid it.) Here the slave is free from interference, but nevertheless it's clear that the slave is not truly free. After all, they remain a slave.
So, Pettit says, this shows the liberal conception of freedom is wrong. Even if a person is not interfered with, they might still be unfree, the liberal would have to agree.
Now, one immediate move the liberal could make is to say that fine, the mere absence of interference is not enough. A person must also have a publicly recognized and adequately enforced right to non-interference. Indeed, liberals have long argued this, so perhaps the problem comes from Berlin oversimplifying the liberal conception of freedom.
Pettit responds by asserting that what makes the slave unfree is that they are dominated by the master. The master has the power and right to arbitrarily interfere with the slave at will and may do so with impunity. Even though the master never does so--indeed, the slave reliably and correctly predicts the master never will--the slave remains unfree because the master has this power and right. So Pettit claims that to be free requires that a person not be subject to such domination.
That seems fine, but what Pettit does with this idea is bizarre. He rightly recognizes that to be dominated by others means one is unfree. People should not be subject to the arbitrary inference of others. Fair enough. But then he ends up defending a philosophy which allows and authorizes a very high degree of interference with everyone, provided that this ioterference comes about the right way and thus doesn't qualify as "arbitrary". (In more recent writings, he modifies the definition of domination slightly, but the details don't concern us here.) What's weird about this is that Pettit realized that domination is bad, but then forgot that what makes domination bad is not merely the arbitrariness/risk of interference, but the interference itself. Pettit is right that the slave with the kindly master is still unfree, so he is right that mere non-interference is not sufficient to make one free. But somehow Pettit moves from "non-interference is not sufficient to make one free" to "non-interference isn't even necessary to make one free". What he should have said is that a person is free when they A) are not interfered with (in certain ways) and B) they are not subject to arbitrary, at-will interference from others, because they have adequately protected and recognized rights against such interference. What Pettit instead says is that a person is free even if 1. they are continuously interfered with in all sorts of ways provided 2. the interference comes about the right way, through a process which somehow pays attention to and tracks their interests, and in which they have a voice.
It's as if someone said, you know, it sucks when you go outside and could get bitten by mosquitos at any time. So what would be better is if we get placed on a regular, scientific, predictable schedule of government-mandated mosquito bites and you get to vote on when they happen.
Pettit thinks democratic processes which engage in frequent interference are fine, and count as non-arbitrary, provided that individual citizens have sufficient ability to participate in the government, the citizens can contest the laws, the government listens to citizens, and there is sufficient division of power such that no one subset of society continuously rules over others. (Again, the details are more complex, but they won't concern us.)
But this brings us to the central problem with democratic freedom. As an individual, except in special cases (such as you are Joe Biden), you have almost no power over what happens in democracy. Democracy does what it does regardless of whether you vote one way, the other way, or not at all; whether you advocate one thing, the other thing, or nothing; whether you protest or stay home; etc. Democratic participation rights are almost always ineffective. From the standpoint of every normal individual in a democracy, democracy is a process by which everyone else can arbitrarily do as they please to do you with near impunity. Giving you the right to participate in this system as an equal usually--except in extremely unusual situations--provides you with basically no protection against what others decide and gives you basically no influence on what happens. It's you with your bucket against the incoming tsunami.
This is one reason why I find certain democratic theorists' views so bizarre. Indeed, they are so bizarre I have a hard time believing democratic theorists sincerely believe what they say. Some democratic theorists downplay individual liberal freedoms in favor of collective participation rights. They claim the latter are more important and valuable than the former. But this means they favor situations in which everyone is powerless as individuals over situations in which each of us has extensive rights of control over ourselves and very little control over others. The only way you could favor that is if you simply don't care about freedom and instead love power and authority. To lionize so-called "democratic freedoms" over liberal freedoms is really to say that freedom doesn't matter.