The Hakka are one of the major subgroups of the Han Chinese. Han is the majority ethnicity, and when someone says "he's ethnically Chinese", that usually means Han.
I'm going to do this as multiple comments so I don't lose my place.
Hakkas are Han, but they're a subgroup and in many ways a distinct ethnicity themselves. They differ from other Han groups in language, culture, dress, food and history.
Origins
The precise origins of the Hakka are not known. There are multiple theories, but the historical records aren't always clear, and there's the added problem that if someone is Hakka now, how far back can you trace ancestry and still be Hakka (or any label, really). We have family histories that place ancestors of Hakkas in the far north, but whether they were there as a distinct group at the time is unlikely to ever have a clear answer. We have records of multiple migrations of the ancestors of modern Hakka going back quite a ways, and so people will say that those were Hakka, but it's clear that at least culturally and linguistically as well as historically, the notion of being Hakka is more recent.
The most widely accepted origin is the Central Plains (中原) region of China. This is actually not really surprising. If there's been one overarching trend in China's history for the past few thousand years it's that people go south. The majority of ethnic groups found in South China and Southeast Asia can trace their origins further north into China than they're found today.
A great many historical figures have been Hakka, Deng Xiaoping being one notable example, and Hong Xiuquan who lead the Taiping Heavenly Kingdom in the civil war known as the Taiping Rebellion was also Hakka. Their history has generally not been so positive, at least as far as interactions with other groups.
The modern name 客家 hak-kâ is often translated as "Guest people", but there's actually some debate about whether this is the actual etymology or not. I can't find the paper at the moment but there's a scholar who's provided some indication that that is not a correct translation, and that it should be something else. However this translation is reflected in the Chinese characters and has caused them no amount of trouble in the past few centuries.
Location
These days the Hakka are found primarily in the provinces of Guangdong and Fujian in China, as well as in smaller numbers throughout the south, and in Taiwan, mostly in the Northwest and on the east coast. In Fujian they're known for their round housing complexes known as tulou 土樓, and throughout the region they're known for their style of dress, which tends to be dark blue and simple for men, and bright floral fabric for women.
They also live in large numbers in diaspora communities throughout the world. I speak Hakka and so I meet a lot of Hakka people and it's no uncommon to find them from Malaysia, Singapore, East Timor, Mauritania, Canada... Pretty much anywhere that's ever had a Chinese community will also have Hakka people.
History
Conflict has often followed the Hakka in their migrations. There have been many instances where conflict has come about due to issues that developed between them as the new migrants and the populations who were already living in the area.
One of the most notable instances of trouble was the Punti-Hakka Clan Wars from 1855-1867, the very name of which expresses their other-ness. Punti is 本地 "original inhabitants" and refers to the Cantonese (Yue) people. Long story short, as the Qing was slowly replacing the Ming as the ruling dynasty, populations in the South fell due to war and migrations, and when the emperor tried to get people to move in to the area to repopulate, it was the Hakka who did the moving. This eventually caused strain as the locals who'd already been there felt resources become more scarce, and eventually this led to armed conflict between the Hakka and the Yue (Penti).
Similar conflict has also occurred in diaspora communities, often based on a similar situation of conflict over resources. Malaysia had significant conflict following the lines of the Punti-Hakka Wars happening in China.
Language
Around 1500 years ago there was no Cantonese, Mandarin, Wu, Hakka languages. There was instead Middle Chinese, from which all of these originated. Hakka is a Sinitic language, similar to Cantonese or Mandarin as it has the same origins. If you heard Hakka sea.
Education
The Hakka historically tended to be more progressive, more open to outside ideas, and generally more educated. During the time of the imperial examination system, Hakkas had a high rate of passing, and among Hakka communities literacy was often many many many times greater than the average. Their openness to outside idea is in large part tied to the Taiping Rebellion as well, as a great many members of the Hakka community at that time were more receptive to missionaries than other Chinese.