Okay, this is a MASSIVE question. And there have been similar questions on here answered before, but this is my own take on it:
I divide antisemitism into religious antisemitism, cultural antisemitism and ethnic/racial antisemitism. Some forms of antisemitism are just based over religious issues, some ethnic, many a mix of both. In some ways, modern antisemitism exists purely because it always has.
Antisemitism has been prevalent across many societies and cultures, from Christian, Islamic and other societies. You can see antisemitism in 19th century books, 20th century genocide and 1st century massacres.
To understand antisemitism, you have to understand the idea of the 'other', and why discrimination and hatred exists against 'the other'. Antisemitism is a form of religious and/or ethnic based othering, a process which identifies people, beliefs and ideas different to your own or mainstream society as other. It splits a society into 'us vs them', with us meaning those who are the same to you! and them being those who are different to you. The 'us' is, of course, seen as the correct, moral or right position, which is why othering can be used to reinforce discriminatory beliefs or ideals, and be used to support or create a system based on subordination and domination. The use of othering reinforces ethnocentricity, nationalism, racism and homophobia, to name just a few.
Othering can also be used to build a communal identity, or strengthen one. This can be a conscious (for example, political, through the use of propaganda) or subconscious decision. Do you feel a certain sense of camaraderie with people who support the same football team as you, people who like the same movies as you and people who are born in your local area? Partly, that is through being able to relate- someone who enjoys the same films as you will be able to discuss characters and plot points with you, meaning you can form a closer bond through meaningful discussions- but also because there is an us, and, an implied them.
So although you feel camaraderie for supporters of your favourite football team, do you feel the same way about all sentient creatures? Probably not. The togetherness, the sense of belonging, that is born out of there being an us, and for humans, it is difficult to turn the idea of sentient beings into an us vs. them thing, meaning we do not feel the pride, happiness etc we get from being an us, as we have not created the idea of a them. For many people, the idea of an us relies on the existence of a them, as some people think the idea of Self relies on the existence of Others.
Within a society, having an Other, having a them, will strengthen bonds, although this is often subconscious. It's often easy to see in the modern world. After 9/11, there was a definite surge in patriotism. There was a feeling of togetherness, the US vs. terrorists, or, even, the US vs. the world. Having the idea of an us and a them (terrorists) reinforced the ideas of shared cultural values between American citizens. It strengthened bonds within the nation.
So there are some basic reasons for why othering is useful- and why creating a minority Other is useful. Within modern societies, there are many others, from Irish Travellers in the UK to the Roma in Eastern Europe to immigrants in South Africa. There are also complex reasons for why those groups are or continue to he the Other, with often the historical othering causing socioeconomic and institutional reasons for why the mainstream society continues to other that community.
Like I said, you can divide it into religious, cultural or ethnic antisemitism. Religious antisemitism would be against followers of Judaism specifically. Cultural antisemitism would be against those raised in Jewish culture (whatever that is), stereotyping Jewish culture into eg being a culture of tight people, bankers or the people controlling the world secretly. Ethnic or racial antisemitism would be against Jews as a whole- born of Jewish heritage, regardless of religion, and against followers of Judaism. Often, antisemitic beliefs combine all three. Someone who believes Jews are miserly penny pinchers who are born or made Christ-killers and who hate Jesus is saying that Jews are born as 'Christ-killers', that they are to be hated both for being born that way and for following a religion which hates Jesus and who have a penny pinching culture and is religiously, culturally and ethnically antisemitic.
If you have someone who is solely culturally antisemitic, then they would believe by ridding a person of 'Jewish culture' or by subverting or changing 'Jewish culture', someone born to eg Jewish parents (but, let's say, removed straight at birth to good parents from the 'right' ethnic, social and religious group) would not be deserving of antisemitism. This is mostly combined with religious antisemitism, as religion is an important and unifying part of Jewish cultures.
If you have someone who is religiously antisemitic, then they would believe that by converting a follower of Judaism to the 'right' religion, whatever that is, they would not be deserving of antisemitism. This is often combined with ethnic antisemitism due to how important religion is in Jewish cultures, communities and lives, and many people see ethnicity and religion as going hand in hand.
Racial antisemitism would be saying anyone who is born Jewish- depending on the antisemite, this could be having any Jewish ancestor, being born to a Jewish mother, born to a family with Jewish grandparents etc- or is now Jewish (through conversion) is deserving of antisemitism. Reasonings for racial antisemitism include seeing Jewish people as being one race which is inferior, and often combines religious and cultural antisemitism as well.
Within each type of antisemitism, there are many antisemitic canards (and many crossover between the types as well.
fyi, you'll find more some related posts in the FAQ