The BBC recounts by means of interviews the experience of a few ambassadors in these matters.
"I was called by the foreign ministry and was told 'We need to see you immediately,'" Mr Casson [former UK ambassador to Egypt] tells the BBC. "The first thing they said was, 'We are not summoning you, but we are going to tell the press we are summoning you. If it had been a summoning, we would have sent a formal diplomatic note summoning you.'" This is the way things normally work in a summoning - a formal, polite, diplomatic note is sent to the relevant country's embassy asking - but not really demanding - its representative to attend a meeting at the foreign ministry, or its equivalent. The medium of the summoning is the message, Mr Casson says. "The main thing is that it is a piece of diplomatic theatre and everybody understands their role, and acts their role," Mr Casson, who was in Cairo between 2014 and 2018, says. In London, the drama can involve being made to wait in the grand surroundings of the Foreign and Commonwealth Office, to understand the seriousness of the occasion. [...] Mr Fujisaki, a former Japanese ambassador to the United Nations, also served as its representative in Washington between 2008 and 2012. In December 2009, he was summoned by US Secretary of State Hillary Clinton. This was notable for two reasons: this is not something the US state department does often, and Japan is a close ally of the US, so an unlikely subject of a summoning. The Washington Post reports Mrs Clinton was "blunt, if diplomatic" with Mr Fujisaki. [...] Although Mr Fujisaki was widely reported to have been summoned, he disputes the use of the word - and says no-one in the diplomatic service uses it. "A summoning sounds like you are obligated to be there, it's more legal," he says. "In this case, the government is asking you to be there, it's a request." Mr Fujisaki says the ambassador must speak to his home government so that they do not learn of the summoning through the media. The home government would then brief the ambassador on the most appropriate way to respond and what points they should make. Ambassadors can be summoned in inconvenient circumstances - as when Israel's foreign ministry called on 12 countries' representatives on 25 December 2016, when many were away for Christmas. So was making Mr Fujisaki travel across a city in a snowstorm a way of inconveniencing him, and reinforcing the state department's unhappiness? He says not. [...] Mr Bocka, the Albanian ambassador in Belgrade since 2014, is in an unusual position: he once refused a summoning by his host country. Four months into his ambassadorship, Albania's football team travelled to Serbia for a Euro 2016 qualifying match. About 40 minutes in, the match was stopped when a drone flew overhead carrying an Albanian nationalist flag. A brawl broke out, and the match was abandoned. Soon afterwards, Mr Bocka received a demarche from Serbia's foreign ministry - a polite request for him to attend, rather than an order. "This is the most common way diplomacy is done," Mr Bocka tells the BBC. "You are asked to go, normally to the director, and you have to hear what they have to say, and at the same time, you have the right to express your position." A year later, it was a different story. Before the return match in Albania's capital Tirana, the Serbian team bus was pelted with stones and a window was cracked. Cue the summoning, and a cranking-up of diplomatic pressure. "They asked me to go immediately to the foreign affairs ministry but I said no," Mr Bocka says. He did not feel the stoning of a bus merited his summoning. "I knew what had happened: nothing! It was 22:00, I said 'maybe tomorrow', but they wanted to have this story on the TV, so I said 'No, I'm not coming'. I told them I wouldn't play this game." Mr Bocka acknowledges how rare it is for an ambassador to reject a summoning by his host country, and the refusal was not without consequences. For several months afterwards, he was disinvited from foreign affairs ministry events, even while his staff continued to work unhindered. But he stayed in his job, and four years later, is on good terms with his hosts again. "Our relations were damaged for a moment," he says. "But now? It's OK."
TLDR: it's usually a fairly polite affair, but it can involve inconveniencing the ambassador on purpose.
Of course, if the ambassador turns out too unpleasant in the judgement of the host, they can be declared persona non-grata and get expelled etc. Although that tends to happen more because of statements to the press, or other deeds, rather than direct exchanges, IIRC. (And of course, if war breaks out or just relations get really bad short of war, the same can happen even if no breaches of protocol occur.)
Or if you prefer a more summarizing (but less exemplifying) take:
The ‘summoning’ of an ambassador is an age-old diplomatic tool. It means that the receiving State sends a note verbale to the relevant country’s embassy requesting the ambassador to attend a meeting at the foreign ministry usually to express displeasure over actions or a policy of the sending State. The modality of the meeting may vary. It has been said that ‘sometimes it can be a casual conversation in comfortable chairs. Other times, a formal encounter across a table. If the row is serious, it can even be what’s called “a meeting without coffee” when chairs are removed from the room, the ambassador is forced to remain standing, and a formal diplomatic reprimand is read out and handed over in text form, known in the trade as a “note verbale”.
Interestingly perhaps, in German practice, with whom the ambassador is summoned or invited to speak to, depends on the 'severity' of the matter:
In German practice, there are various degrees of summoning. At its most mild, the ambassador is requested to attend a meeting with the director or a representative of the relevant department in the Federal Foreign Office. To escalate things, the conversation is conducted by a State Secretary. If things are really serious, the ambassador may be hauled before the Federal Foreign Minister. For example, in October 2013, Federal Foreign Minister Guido Westerwelle summoned the US ambassador to Germany over allegations that the United States has been tapping Chancellor Angela Merkel’s mobile phone.
A related piece in the same venue adds a few more details (mostly) about the German practice:
The Vienna Convention on Diplomatic Relations does not contain any provisions on the summoning of ambassador. Summons is, however, a well-established practice of diplomatic relations. [...] Heads of mission are, as a rule, summoned to the Ministry for Foreign Affairs and not to other government departments, the chancellery, or prime minister’s office. Heads of mission are usually summoned at the behest of the Foreign Minister of the receiving State. In terms of diplomatic protocol, summons are extended by Deputy Foreign Ministers, State Secretaries, and the directors of Foreign Ministry departments. Being summoned is part of the functions of the head of a diplomatic mission. However, it is the sovereign decision of the sending State how the functions of a head of mission are exercised. There is no obligation under international law to comply with a summons, but failure to do so may prompt the receiving State to declare the head of mission persona non grata.
And, often enough, a public statement is then made as to what happened, e.g.:
On 28 May 2020, a Federal Foreign Office spokesperson issued a statement which read in part: “State Secretary of the Federal Foreign Office Miguel Berger invited Sergey J. Nechajev, the Ambassador of the Russian Federation, to the Federal Foreign Office for talks today. On behalf of the Federal Government, Mr Berger condemned the hacker attack on the German Bundestag in the strongest possible terms. The Russian Ambassador was informed, with reference to the arrest warrant issued by the Investigating Magistrate of the Federal Court of Justice at the request of the Federal Public Prosecutor’s Office on 29 April 2020 against Russian national Dmitri Badin, that the Federal Government will call in Brussels for the EU’s cyber sanctions regime to be invoked with respect to those responsible for the attack on the German Bundestag, including Mr Badin. […]
I'm not sure if the German side (officially) publicized anything of the Russian response, or if they usually just use the occasion to [re]state their own position. Anyhow, the mass media will probably report what the other side said.
Another piece about how these things were handled in Russia (in 2021):
The American ambassador being summoned to our MID [the Russian Foreign Affairs Ministry] is a common practice. They summon him for consultations, to clarify some positions that are being expressed by his country’s leadership. [...] It would be more appropriate to say invited than “summoned.” The process goes something like this: a document requesting a visit from the ambassador is sent from the MID to the embassy. The time is agreed upon through the usual channel, based on the MID’s schedule and the ambassador’s schedule. [...] [During the meeting], the MID will ask the ambassador clarifying questions, however, he isn’t obliged to answer everything immediately. If any questions are uncomfortable for him for whatever reason, no one will grill him — they simply have no right to interrogate him. He can take a time-out and consult with the central apparatus — in [the U.S.’s] case, the State Department. If he doesn’t want to answer the question directly, he’ll give a vague response. [...] Previously, there were instances where they asked the ambassador an uncomfortable question and he avoided answering it by saying “I need to think” [or] “I’m not prepared to answer,” or he hit back with platitudes. A relatively new trend in international communications is that everything is said directly.
A 2010 FP piece summarizing in the abstract, but I get the impression based on mostly how China, the USSR, and perhaps the US dealt with these:
These summonses tend to be highly choreographed affairs. Imagine that one country — say, China — is angry at the United States for selling arms to, say, Taiwan. China’s foreign minister decides to summon the U.S. envoy in Beijing, releasing a statement detailing why the sale is unacceptable. When the ambassador shows up at the Foreign Ministry, a delegation of Chinese officials meets him and explains the country’s grievances, reading from a prepared statement. Then, the U.S. ambassador responds by stating his government’s views. Depending on the circumstance, he might apologize, request further dialogue on the issue, or even reject China’s position outright. Stagecraft is paramount. Generally, a summons will include two to four diplomats on either side. But if an ambassador comes in to meet with a cabinet minister or a massive delegation, she knows she is in real diplomatic hot water. Ministries also carefully engineer the tone of the meeting, from the conciliatory to the very frosty. They decide how strong to make the language they use with the ambassador — sometimes, even, using condescending or discourteous terms. In a rancorous summons in a French-speaking country, for instance, a diplomat might use the "tu" rather than the "vous" form of address as a not-so-subtle slight. (Such lack of decorum, though, is very rare.) Responding ambassadors — who are coached on how best to respond to summonses by their foreign ministries and embassy staff — engineer careful responses in turn. The Soviet Union was notorious for its chilly meetings — literally. U.S. Embassy staffers would show up to find themselves facing a contingent of unsmiling diplomatic officials in a freezing-cold room. Of course, relations between Washington and Moscow weren’t so good back then — and the warmth of the relationship between two countries determines how often, if ever, an ambassador is summoned. (The U.S. ambassador to Switzerland virtually never gets summoned.) The U.S. State Department tends to use summons very sparingly — only to telegraph high-level diplomatic concern. [...]
Alas the rest of the last para just exemplifies what counts as such concern in the US, but details nothing of the US protocol otherwise. (And according the "thanks" note at the end of that article, it's written based on the experiences of "several former ambassadors", which I'm guessing mostly means US ambassadors, since they also thank the US State Department for helping write the piece.)