The United Nations defines genocide in Article II of the Genocide Convention as:

“Any of the following acts committed with intent to destroy, in whole or in part, a national, ethnical, racial or religious group, as such:

• Killing members of the group;

• Causing serious bodily or mental harm to members of the group;

• Deliberately inflicting on the group conditions of life calculated to bring about its physical destruction in whole or in part;

• Imposing measures intended to prevent births within the group;

• Forcibly transferring children of the group to another group.”

https://www.un.org/en/genocideprevention/genocide.shtml

To support the label genocide, some have pointed to Israel’s policies during the current conflict, the rules of engagement in times of war, and the nature of its bombing campaign. They have argued that the number of civilian casualties and the de-facto depopulation of northern Gaza point to genocide and the intentional destruction of the Palestinian people. Learn more here: https://www.aljazeera.com/opinions/2023/11/2/a-genocide-is-under-way-in-palestine

While acknowledging the civilian death toll in Gaza, Israel has maintained that it is engaged in war and that there is no indication of any attempt to eradicate or intentionally harm the Palestinian people as a whole. In fact, Israel argues that its habit of warning civilian populations before attacks, of allowing humanitarian aid to enter conflict areas and of conducting relatively targeted strikes indicates the exact opposite: https://www.haaretz.com/opinion/2023-11-01/ty-article-opinion/.premium/israel-is-not-committing-genocide-in-gaza/0000018b-8785-d055-afbf-b7a75d450000?lts=1700587783609

The term “apartheid” originally referred to a series of laws passed in South Africa between 1948 and the 1990s, institutionalizing racial discrimination and the dominance of the nation's white minority: https://www.sahistory.org.za/article/history-apartheid-south-africa

The United Nations later generalized the term, declaring and formally defining the "Crime of Apartheid" and racial segregation: https://legal.un.org/avl/ha/cspca/cspca.html

Critics of Israel have indeed used the label apartheid when referring to Israel's relation to the Palestinians. See for example: https://www.amnesty.org/en/wp-content/uploads/2022/02/MDE1552342022ENGLISH.pdf

However, the accuracy of the term is contested: https://www.standwithus.com/factsheet-apartheid-slander

There appears to be no formal definition of the word “terrorism” in international law. However, many nations have adopted their own definitions, and several United Nations resolutions and conventions refer to terrorism and characterize it. For example, the United Nations Security Council Resolution on International Cooperation in the Fight Against Terrorism defines it as “criminal acts, including against civilians, committed with the intent to cause death or serious bodily injury, or taking of hostages, with the purpose to provoke a state of terror in the general public.”

The resolution also declares that terrorist acts "...are under no circumstances justifiable by considerations of a political, philosophical, ideological, racial, ethnic, religious or other similar nature.” Learn more here: https://digitallibrary.un.org/record/532676?ln=en

Hamas began as an offshoot of the Muslim Brotherhood, which you can learn more about here: https://www.cfr.org/backgrounder/egypts-muslim-brotherhood#:~:text=Founded%20in%20Egypt%20in%20the,Spring%20in%20the%20early%202010s.

However, in 1988 Hamas published a charter calling for violent eradication of Israel and of the Jews living within its borders and positioned itself as a religiously-oriented armed resistance organization. You can read it here: https://irp.fas.org/world/para/docs/880818a.htm

Notably, they updated their charter in 2017 and removed some of the calls for violence against Jews and Israel. You can read that one here: https://www.middleeasteye.net/news/hamas-2017-document-full

Hamas is designated a terrorist organization by the US and other countries because, since its inception, it has directed attacks against civilian targets. These include suicide bombings, rocket attacks and kidnappings. See here: https://www.state.gov/foreign-terrorist-organizations/

Some, notably Turkey, call Hamas not a terrorist organization, but a “liberation group.” You can read about that here: https://www.reuters.com/world/middle-east/turkeys-erdogan-says-hamas-is-not-terrorist-organisation-2023-10-25/

Al Jazeera calls Hamas a “Palestinian armed group,” and explains it’s history here: https://www.aljazeera.com/news/2023/10/8/what-is-the-group-hamas-a-simple-guide-tothe-palestinian-group

Many observers of the current Israel-Gaza war have called for an immediate ceasefire, on humanitarian grounds.

Israel has repeatedly argued against the ceasefire on several grounds. They observe that Hamas appears unrepentant and entirely committed to continued attacks on Israeli civilians. Consequently, the claim is that Hamas will use any ceasefire to re-arm and entrench, only prolonging the war. See here: https://twitter.com/AJCGlobal/status/1720068833028059335

Second, Israel argues that Hamas has a history of violating ceasefires and consequently a ceasefire with them cannot be negotiated in good faith: https://www.jpost.com/opinion/hadar-goldin-is-the-victim-of-a-humanitarian-cease-fire-564520

Finally, Israel maintains that a major goal of the current operation is the rescue of the hundreds of Israeli hostages still held by Hamas. They fear that a ceasefire would allow Hamas to move the hostages with impunity and void any intelligence Israel currently has regarding their whereabouts: https://www.ajc.org/news/6-reasons-why-an-israel-hamas-ceasefire-is-not-the-right-path-forward#:~:text=Israel%20has%20been%20firm%20that,on%20the%20hostages%27%20current%20whereabouts

Supporters of the ceasefire argue that the humanitarian toll is unbearable and that no strategic or military goal, even if legitimate, can justify the level of harm being done to Gaza and its population. Ceasefire proponents also claim that the most effective way to ensure the release of the Israeli hostages is by means of diplomatic negotiations. They argue that the military operation actually puts the hostages at greater risk and reduces the chances of their safe release. https://www.amnesty.org/en/petition/demand-a-ceasefire-by-all-parties-to-end-civilian-suffering/

For context, in 2005 Israel withdrew its troops from Gaza, leaving it under the control of the Palestinian Authority (PA), the Palestinian governing body established by the 1993-1995 Oslo Peace Accords: https://www.npr.org/2005/09/12/4841877/last-israeli-troops-exit-gaza-strip

However, in 2007, Hamas wrested control of Gaza from the PA: https://web.archive.org/web/20080907211849/http://www.pchrgaza.org/files/PressR/English/2007/news/54-2007.html https://irp.fas.org/world/para/docs/880818a.htm

Consequently Israel and Egypt, the two countries bordering Gaza, imposed a joint blockade on the region with the stated goal of keeping weapons out of the hands of Hamas, and with the occasional support of the Palestinian Authority: https://www.aljazeera.com/news/2014/12/12/abbas-supports-egypt-action-on-gaza-tunnels/

The blockade has been a significant source of tension between Israel and Hamas, driving a long and violent series of conflicts including a variety of attacks on Israel by Hamas and Israeli retaliations in the form of air strikes and ground incursions: https://www.britannica.com/place/Gaza-Strip/Blockade

While Israel has generally allowed basic necessities into Gaza, the blockade has been accused of causing significant hardship and humanitarian crises within the Gaza Strip: https://www.theguardian.com/world/2010/nov/30/israel-accused-over-gaza-blockade

Following the October 7th attack on Israel by Hamas, Israel imposed a total blockade disallowing the entry of all goods into Gaza. The total blockade lasted until October 18th when restrictions were eased to allow entry of humanitarian aid into the region: https://www.reuters.com/world/middle-east/israel-will-not-allow-gaza-supplies-israel-not-block-egypt-pm-2023-10-18/

The current blockade has drawn significant criticism for creating shortages of basic necessities in Gaza. However Israel has declared that it will not be lifted until the hostages abducted by Hamas are returned to Israel safely: https://www.bbc.com/news/world-middle-east-67051292