Are the two cells that are derived from one cell, ‘twin sisters’ or a ‘mother and a daughter’? In other words, can a cell really be divided to live an "immortal" life or is cell reproduction the reality of cell life that accounts cell age difference between two successive generations, cell aging-associated senescence in cell reproduction, and cell line survival after the death of old cells.
First of all, in eukaryotes (as far as I'm aware), older cells can be distinguished from younger cells due to telomere shortening, so there is an ageing process. HeLa cells mentioned by @Gary Chou have a more active telomerase which mitigates telomere shortening, allowing cells to continue to divide indefinitely. I think it's a very interesting question whether this immortality eliminates any sense of "older" and "younger" cells from in HeLa cell culture. I suspect not, but I am not sure on the specifics.
In general this question will depend on how an organism divides (we're going to talk single-celled organisms from now on). For example, the yeast Saccharomyces cerevisiae is a budding yeast, which means that it divides by budding a smaller cell off of the larger cell. This clearly allows for an interpretation of "mother cells" and smaller "daughter cells." Indeed, as lots of research on ageing in S. cerevisiae has shown, beyond being larger, the mother cell undergoes replicative ageing: over time the mother cell is less capable of growth and division--she gets old!
Perhaps the most interesting case of all (and maybe the one you had in mind) is when the organism divides symmetrically, like the bacteria, which undergo binary fission. For decades, as far as scientists could tell, the two cells that were the product of one bacterium dividing looked identical, and all measurements seemed to indicate that the cells were "sisters," i.e. exactly the same, both newborn. So if in the right conditions bacteria could continue to divide indefinitely, it seemed that bacteria were immortal, since each division reset the ageing clock for each bacterium.
However, a seminal study in 2005 (1) changed this view. The researchers took time-lapse microscope movies of a colony of Escherichia coli from a single cell. E. coli is rod-shaped, so when it divides it has to synthesize two new ends or poles, one for each cell, so just after division each cell has a "new pole" that was made in the process of that division and an "old pole" which had originated before. In this study Stewart et al. kept track of the old poles in the growing colony, and they defined the age of a cell as the age of its old pole. They found that older cells (i.e. cells with older poles) displayed replicative ageing, just like yeast! They grew more slowly and were less capable of dividing. This was examined rigorously by Wang et al. (2) in a "mother machine" in which bacteria were able to grow in constant nutrient conditions (by flowing fresh media across them) in dead-end channels. The cell at the dead-end of the channel always inherited the old pole, and so was a "mother" to all future cells. They found that, while a cell line was able to grow for very long periods of time completely stably, after hundreds of generations there was an increased likelihood for the cells to display signs of stress. They attributed to the accumulation of toxic factors in the mother cell over time. So even in cells dividing by binary fission, there appears to be some difference between the two cells arising from the division, depending on how old the pole that they inherit is.
These effects in bacteria are small, and require watching many generations to reveal. Even in light of these results many researchers will still call the "mother" cell the cell about to divide and the "daughter cells" the cells that result from division, implying that there is no difference between "daughters." Stewart et al. claim that on the basis of their work this conception of bacterial division must be revised, and that instead of division producing two identical daughters,
the old pole cell should be considered an aging parent repeatedly producing rejuvenated offspring.
Furthermore, they add a nice philosophical touch to their findings:
These results suggest that no life strategy is immune to the effects of aging, and therefore immortality may be either too costly or mechanistically impossible in natural organisms.