The X-shaped picture, explained
The classic textbook chromosome looks like an X. That shape appears only at a specific moment: just before a cell divides, after the DNA has already been copied. At that point the chromosome consists of two identical halves, called sister chromatids, joined together. Each chromatid is a complete copy of the same DNA molecule, made during DNA replication.
The two sister chromatids are pinched together at a constricted spot — the centromere. This is the chromosome's waist, and its position is consistent for each chromosome, which is one of the features used to tell chromosomes apart. Before replication, or after the sisters separate, a chromosome is just a single rod with one centromere — no X at all.
Caps at the ends: telomeres
The very tips of each chromosome are protected by telomeres — repetitive stretches of DNA that act like the plastic caps on the ends of shoelaces. They do not carry genes; their job is to keep the natural ends of the molecule from fraying or accidentally sticking to other chromosomes, which the cell would otherwise mistake for dangerous breaks.
Telomeres also buffer a quirk of copying: each round of DNA replication leaves the very end slightly incomplete, so chromosomes shorten a little with each division. Because telomeres are made of disposable repeats, this trimming nibbles harmless caps rather than precious genes — at least for a long time.