The 'Apiru were basically nomadic raiding groups. It appears to have been a term from Ugaritic 'pr and Egyptian 'prw. According to Grabbe's Ancient Israel: what do we know and how do we know it, the term 'Apiru is

[not] an ethnic term, but a social designation: the word seems to mean someone outside the social system or an outlaw, or a refugee or migrant (p. 48).

Van de Mieroop (2007) describes the 'Apiru as

outcasts from society who sought refuge in territories who were difficult to control and who lived outside the structures of the states (p. 198-199).

Therefore 'Apiru isn't an ethnic term, like Hebrews or Sumerians or Egyptians. It's a social designation, like immigrants or citizens or priests. 'Apiru were marginalized people, who lived outside of established societies, and sustained themselves off of those established societies via banditry, raiding, and mercenary work.

'Apiru feature quite prominently in the Amarna letters, where many of those labelled 'Apiru seem to have sold themselves as mercenaries or turned to raiding or stealing. This meant that it became a sort of pejorative term, kind of like "bandits" or "outlaws". 'Apiru may have many origins, and may only be 'Apiru for a short time. For example, we have a text from Idrimi, king of Alalakh (a city in modern Syria) who joins a group of 'Apiru when he becomes a refugee and lived among them for seven years (ANET 557). They were migrants, people on the edge of society with no real "in" to any settled civilization, and had to resort to socially marginal ways of making a living, which included mercenary work and raiding and general banditry.

Matthews (2002) describes the 'Apiru as

stateless people ... [with] only loyalty to their own group (p. 27)

Matthews also makes a note that 'Apiru people were among the Semites who were employed by Egypt to transport stones for building the store city of Pi-Ramesses, as their rations are listed in the Leiden Papyrus 348 (p. 27), so we know they are a marginal group in Egypt as well.

Now, what is the relation between the Hebrews and the 'Apiru? Think of it as a Venn diagram, where all Hebrews are 'Apiru, but not all 'Apiru are Hebrews. The Hebrews according to all the texts we have describing them in this period of their history fit into the definition of 'Apiru: a marginalized group if migrants who force their way into a previously established society. Grabbe goes on to indicate that the word Hebrew is probably derived from 'Apiru, and perhaps this small group of 'Apiru took the term and made an ethnic identity of it (p. 48, citing Na'aman 1986). So the Hebrews were a specific subset of 'Apiru people who reclaimed a territory out of their marginalization.

Sources:

• Grabbe, L. L. (2009). Ancient Israel: What do we know and how do we know it?. London: continuum.

• Matthews, V. H. (2002). A brief history of ancient Israel. Louisville, Ky: Westminster John Knox Press.

• Van de Mieroop, M. (2007). A history of the ancient Near East: Ca. 3000-323 BC. Malden, Mass. Blackwell.

EDIT: added a page number.