Telomeres are "short repeated sequences of DNA (TTAGGG)" that form the very end of the chromosomes. Their function leaps out at you once you appreciate an odd quirk of the DNA copying mechanism: chromosomes cannot be copied all the way to their extreme ends. Some small portion is always unavaoidably left out. (Again, if you want the reason, leave a comment to that effect.) Thus telomeres act as 'buffers' during cell divisions (when chromosomes are copied), making sure that the bit that's left out isn't some crucial gene. And, since the telomeres' extreme ends aren't copied, they tend to become shorter after each replication. (Telomeres also function to protect the ends of the chromosome from being degraded or from "sticking" to another chromosome accidentally.)
Yeast cells, protozoa and other single-celled organisms get around this inevitable shortening by using the enzyme telomerase, which simply regenerates the telomere. In humans, telomerase is also present in germ cells and, to a minor degree, in stem cells.
However, in fully differentiated human cells, telomerase is inactive. Therefore, as a cell divides and divides to make clones of itself, it's telomeres continue get shorter. When they eventually pass a critical threshold of 'shortness', the cells become unable to divide further.
Thus, many single-celled organisms are able to divide indefinitely, whereas most of our cells only divide a certain number of times. Why? This seems to be an anti-cancer mechanism; a protection against the unregulated cell growth that is so typical of a cancer.
In at least 90% of human cancers, telomerase is reactivated. This immediately suggests a potentially fruitful line of research: in a patient suffering from cancer, could telomerase be inactivated again, and would that cure the cancer?