Telomeres are often described as the 'end caps' of chromosomal DNA. They protect the terminal sections of chromosomes from progressive degradation during cell division. Uncapped, the chromosomes would not be able to repeatedly copy their end-regions without errors (due to the so-called End Replication Problem).
As chromosomes are duplicated during cell division, the telomeres become gradually degraded and shorter. At some point, they become so short that the cell can no longer reliably replicate. Human lab-grown cells, for example, tend to be unable to divide more than 50 or so times (ref.).
This phenomenon led to widespread adoption of the view that an organism's telomere length could be used as a reliable measure of how senescent an individual animal is - at least in terms of its DNA integrity at the cellular level.
Note that In the same way that the number of genes which an organism has varies greatly from species to species, (see genome sizeplugin-autotooltip__plain plugin-autotooltip_bigGenome size
The size of the genome (the amount of DNA in an organism's genes) varies enormously from one species to another. The smallest sizes, for viruses, ranges from around 2 thousand base-pairs to over a million.
Humans have about 3 billion (forming around 19,000 genes *) - but some plants have more than) telomere lengths also dramatically vary. From around 300 base-pairs in yeast, to around 10,000 in newborn humans.
New research, published in 2023, has indicated that the implications of telomere lengths could be considerably more complicated than were previously thought.
The studies have shown that longer telomere sections can, in some cases, lead to increased disease risk. (ref.)
A further consideration is that telomere lengths appear to have different implications for different organisms. In mice for example - which have a comparatively short lifespan compared to humans - the telomeres are around 5 times longer. In addition, their telomere lengths tend not to change very much over time. (ref.)
To sum up : It's known that cells which have undergone many divisions tend to feature more DNA mutations, and are therefore more prone to disease and malfunctions. But, the idea of using the length of the telomeres as a measure of what might be called the 'youthful healthiness' of an organism appears to be substantially less reliable than was previously assumed.
The current thinking is that human individuals with moderately degraded telomeres tend to be more healthy than those who have very short or very long ones. More research is needed to determine if there really are any reliable 'rules' for using telomere lengths as an indicator of health and/or age.