It's just called "old age". And yes, old age is displayed on the cellular level, as well as on the macrocellular level.
Basically, when a cell (whether it's a single-celled or multi-cellular animal)gets old, cellular bodies called lysosomes start appearing with more frequency. We're not sure why they start appearing or what function they serve, but they're a clear sign of old age & a clear indication that the cell isn't long for this world.
What's really interesting is that we're not sure how having a high proportion of cells with high lysosome counts relates to the signs of old age that we're familiar with (i.e. skinspots, frailty, etc.), but there must be some connection since there's a 1:1 correlation between the two.
No. Telomeres are structures that appear at the end of chromosomes. They're involved with the replication of DNA when a cell divides. (i.e. When a cell divides, both of the resulting cells get a full copy of the original cell's DNA.)
What you might be referring to is that as an organism ages, telomeres start appearing with more & more errors, thus resulting in errors when the DNA is copied. Again, as far as I know, we're not quite sure why the telomeres of multi-cellular organisms start to display more errors as the overall organism ages.
Keep in mind that it's been 20 years since I got my undergraduate degree in biochemistry; science may have advanced considerably in this area since then.
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