The species
Ursus arctos, or brown bear, consists of all the bears that are not other species of bears. They share common descent, like the members of your family. They share some distinctive alleles (gene variants) or combinations of alleles that set them apart from other bears. They share some distinctive physical and behavioral attributes and, likely, they share an ecological niche that is different from that of other bears. They may eat different food or breed at a different time of year.
In general, the most important unit in evolution and ecology is the population. A species may consist of a single population or of multiple populations. A group of populations my individually go extinct, allowing their particular geographical area to be recolonized by individuals from another nearby population. Although this may result in minor changes in gene frequency, the species remains intact. Even if some individuals die, and even if some populations die out, the species continues to exist.
In the same way, even if a few of our brain cells die or even if we lose an arm or leg, we are still individuals. Even if someone receives a hand transplant, he is still the same person. I think it's fair to say that in evolutionary terms, a species is comparable to an individual.
All of the individuals in a population share a series of characteristics and the potential ability of interbreed with one another. The same is true of the species generally. Occasionally, individuals can successfully breed with individuals from other populations that would normally be considered a separate species based on genetic or ecological characters.
What happens to such progeny is instructive. If they are fertile, either with themselves or with individuals from either parent species, and if the environment allows it, such hybrids may thrive. But if any one of these things is not true, they will die out quickly, reinforcing the boundaries between the two species even though they don't seem to have sharp reproductive boundaries. Generally, a harsh environment causes "stabilizing selection" around a phenotype that is functional if not perfect. In times of plenty, variants may arise. In harsh times, the variants (on the edges) and hybrids (between two species) are more prone to dying out. These are generalizations only. There will be exceptions.
When species are in the process of splitting into two or more separate species of if a hybrid is arising, these distinctions become difficult. But there are likewise situations where it is unclear how to define individuals.
Consider Embryologist Scott Gilbert's comments on twinning in humans.
"In humans, identical twinning can occur as late as day 12 [after conception]. Such twinning produces two individuals with different lives. Even conjoined ("Siamese") twins can have different personalities. Thus, a single individuality is not fixed earlier than day 12."
Developmental Biology 8e Online: Summary
Would we consider a partly split embryo two people or one? Is it truly one individual on day 11 and two on day 12?
I don't think that a single individual type could ever be considered a species by any definition--genetic, ecological, or reproductive (the biological species concept, or BSC). It would be considered an individual with major birth defects.
Even though I am discussing species in very abstract terms, as entities, I think they are quite concrete and real. Yes, they have fuzzy boundaries. But as you have pointed out elsewhere, most entities have fuzzy boundaries at some level--whether atomic, molecular, or cellular. Having fuzzy boundaries doesn't make a dust bunny an abstraction.
Individuals are made of cooperating cells, whose constituent molecules are typically entirely replaced during the lifetime of the individual. In addition, we harbor hundreds of commensal and symbiotic bacteria and fungal species whose combined numbers vastly outnumber our own cells. As individuals, we are cooperatives and communities.