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Every election is a little bit different, and they all change over time, but in the middle nineteenth century the process was basically that you had three guys at the polling place. Their job was to make sure only legal voters voted, which they did by either knowing the voters or getting them to declare they were legal under penalty of perjury. If there's disagreement between the three, then they vote amongst themselves. Depending on the circumstances, there may also be a clerk or two who does the actual paperwork. If any of the judges of election failed to appear, the voters present could nominate their own replacement(s) by acclamation. When the polls close, the same three judges do the math and then submit written returns to the relevant authority. There was also a window during which people present at the polls could dispute those returns, which might result in a do-over. Basically, though, it's down to how much voters trust the local officials. This is often quite a bit, since they would be prominent men within the community.

Then there's the times when it's not enough. The most infamous case of this in the nineteenth century US is probably territorial Kansas. Kansas was opened to white colonization and slavery by the Kansas-Nebraska Act. Officially, it gave the power to the local whites to decide for or against slavery but really it repealed the prior ban on slavery in the territory and threw open the gates. Said gates opened onto Missouri's slave labor camp belt and were opened at the insistence of the enslavers there, who understood their own enslaving to be imperiled if they allowed free territory right next door.

Antislavery whites were not about to take that idly. The law said that the white men on the ground would decide, so they organized to get antislavery men into Kansas. This mainly involved sponsoring emigration. They'd get you a good deal on transport, that kind of thing. Proslavery Missourians believed they had not exerted such effort to secure Kansas for slavery to have it squandered by some damned Yankees who thought this could be settled by something so trifling as voting. They counter-organized, mostly out of Masonic lodges.

Turnabout is fair play, if one is adequately white. That is not what the Missourians, led by Senator David Rice Atchison, had in mind. They organized people not to go to Kansas to stay, but rather to go over the border at election times. They would go by the thousands, well-armed (including with two cannons), and well-lubricated. They had one mission, to bring slavery to Kansas whether Kansans wanted it or not, and they attacked the problem at both ends. Their warm bodies supplied thousands of fraudulent votes -you had to be an actual resident of Kansas to vote legally, which none of them were and few even pretended to be and they used their numbers, angry disposition, and actual guns and knives to keep antislavery men away. Most of them saw the columns of Missourians and didn't even try. Those who did received dire threats credible enough that proslavery actual Kansans were sometimes put off and at least one guy had to take shelter with the judges of election lest he be mobbed and possibly killed.

It gets worse. The one guy who took shelter with the election officials did so in a poorly-attended nothing of an election, Kansas' first. That's for the non-voting delegate to Congress, November 29, 1854. That Congress ended the following January. It would be hard to arrange a less consequential election. Come March, the polls are open for the territorial legislature. That's where Senator Atchison comes across with his cannons. He makes a loud speech at one polling place about how they'll kill every god-damned abolitionist in Kansas if they have to. In a few places, even local proslavery candidates are forced to withdraw in favor of Atchison-approved men. The Missourians crowd the polls and don't let anyone near who they aren't sure of. One guy claims he's proslavery and actually gets hoisted up and carried through a crowd...right up until they see that he's carrying an antislavery ballot. He made it out alive, but didn't get to vote. So that's the first prong: voter intimidation and suppression. It's downhill from here.

Atchison and company did not cross the border with their bowie knives, guns, cannons, and all that alcohol just to get drunk in the jurisdiction next door. The judges of election could still refuse to take their votes, which would make the whole exercise pointless to them. Ahead of the election, various known antislavery judges are warned to stay home. Those who do leave it open for the crowd of Missourians to appoint their replacements. That goes a little ways farther. You only need to get one guy not to show up to get a second proslavery man on the panel and fix things up. But sometimes there are two antislavery guys or one guy who doesn't care much either way but thinks this is all a humongous hot heaping helping of bullshit.

To fix a "problem" like that, you need those guns and knives. Stuff like this happens on a small scale, at least, in every polling place we have information on. Probably the most outrageous case is in the Second District, Bloomington precinct. There, three to seven hundred Missourians arrived and found that they had sort of been expected. The judges located their polls in a log cabin, which they shut up except for a window to take votes through. They might have been armed, but the witness who tells us so also tells us there were no Missourians present. Between fifteen minutes and an hour of unmolested voting took place before Missouri rolled in.

They came under the leadership of Samuel J. Jones, later a prosalvery sheriff in Kansas and later still shot in the back by (probably) antislavery persons unknown (to us). Jones and his men found out that the judges of election took their instructions from the governor seriously: only actual Kansans voting, thank you. They didn't know any of these men, but the procedure was for them to swear out an oath and then be counted. According to another Samuel Jones who is not the sheriff but was present there that day because people in the past hate historians with transcendent fire, future sheriff Jones said:

replied that they had come up from Missouri to vote, and they would vote, and they intended to make Kansas a slave-State. They [the Missourians] also stated that no man should vote there that day that would take the oath.

Jones (the sheriff) was speaking to the poll workers through their window at this point. A few unaffiliated sorts had been hanging around the polls -this was normal at the time, election day was sort of a general holiday and people would gather to socialize and drink- and they tried to vote. The Missourians hauled them back by force. Then came the problem of those recalcitrant judges. Jones made them an offer with a low probability of denial:

that if we did not do so [resign] they would tear the house down and kill us.

They meant it. On hearing the judges' refusal, the mob rushed the window, breaking it and its sash. Also they reminded the judges that they were armed. As the clerk, William Jessee, put it:

There were, I should think, a dozen pistols cocked and pointed to the judges at the window, and they swore they would blow their brains out if they did not receive those votes or resign. I did not count the number of times these pistols were presented, but I should think it was from eight to a dozen times.

That did not do the job, so the Missourians stuck long logs under the corners of the house, shoved other logs under those, and tried to lever it off its foundation. They gave it enough of a heave that the building lurched up a bit.

Paris Ellison, one of the judges, did the math:

Mr. Ellison, gathered up the ballot-box and rushed to the door, and said that if we did not close the polls there would be one hundred shots fired in here in less than fifteen minutes, and we would all be killed. He opened the door and ran out into the crowd, taking the ballot-box with him, and hurrahed for Missouri.

This is a key moment. It seems that the cabin had been locked against the Missourians. Eillison left the door wide open for them and in they rushed, Jones in the lead.

with revolvers drawn and bowie-knives in their hands. They now filled the house; and Jones, one of the first who came in, requested us to resign; that if we did not resign they would kill us. Jones drew from his pocket his watch, and gave us five minutes in which to resign or die.

That's judge of the election Harrison Burson. Nathaniel Ramsay, the third judge present, estimated six to eight Missourians

had their revolvers ready to shoot and their knives in their hands ready to stab.

Five minutes came and went. Jones gave them one more minute to pull out. Word arrived that a John Wakefield -who is actually the guy who took shelter with poll workers last election- wanted to speak to Burson about the situation. He promised to answer Jones on returning. Burson didn't know, apparently, that Wakefield had been in the hands of the mob for a while. According to him,

A man by the name of G.W. Ward came to me and said, “Judge Wakefield, if you have any influence over those judges, you go and tell them to let the men vote.” I told him that I never advised men to do wrong.

George Ward, incidentally, is the one witness who said there were absolutely no Missourians present at the polls that day. He was also the guy running on the proslavery ticket. Small world.

Per Wakefield,

A man then, that I have since understood was named Jackson, got on to a small log and made a very inflammatory speech. He told them to divide themselves into companies and time a white ribbon in the button-holes of their coats, that they might know their party from the abolitionists